Multidimensional Hypergeometric Integrals, Quantum Differential Equations, and Integrable Systems

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The development of mathematical analysis in the eighteenth century led to the development of new special functions in addition to polynomial functions and trigonometric functions. In particular, the famous hypergeometric function was introduced and studied in the eighteenth century by Leonhard Euler and then by the leading mathematicians of their time: Gauss, Jacobi, Kummer, Fuchs, Riemann, Schwarz, and Klein. Modern versions of that function appear in different mathematical and physical theories (such as representation theory, algebraic geometry, gauge theory, statistical mechanics) and are considered in these theories from different points of view. The goal of this project is to develop a unified analysis and geometry of modern multidimensional hypergeometric functions with applications to the above theories. It will lead to a better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them. The project includes the training of graduate students.This project involves research on representations of quantum groups, quantum cohomology and associated quantum differential equations, KZ type differential and difference equations, algebras of Hamiltonians of quantum integrable systems, Bethe ansatz method, and hypergeometric functions. In particular the principal investigator plans to: 1) construct integrable representations for solutions of the quantum differential equations and different types of differential and difference KZ equations with applications to the three dimensional mirror symmetry; 2) study the reduction of these equations and their solutions modulo a prime number; 3) study of the spectrum of commuting Hamiltonians in the associated quantum integrable systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.