Critical Points of Master Functions, Hypergeometric Integrals of Arrangements, and Quantum Integrable Systems

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The hypergeometric function was introduced and studied in 18th century by Leonhard Euler. Modern versions of that function appear in different mathematical and physical theories (like 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 better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them.This project involves research on representations of quantum groups, algebras of Hamiltonians of quantum integrable systems, quantum cohomology and associated quantum differential equations, Frobenius structures, Bethe ansatz method, theory of arrangements of hyperplanes, singularity theory of critical points of functions. In particular the principal investigator plans to: 1) construct q-hypergeometric solutions of the equivariant quantum differential equation for the cotangent bundle of a partial flag variety; 2) identify the quantum cohomology algebra of the cotangent bundle of a partial flag variety with the algebra of functions on the critical set of the associated hypergeometric (or q-hypergeometric) master function; 3). use hypergeometric solutions of qKZB equations to define an action of elliptic dynamical quantum groups on elliptic equivariant cohomology of partial flag varieties; 4) find a potential for a KZ-type connection and a Frobenius-like structure on the base of the KZ-type connections; 5) develop a relation between the critical set of master functions associated with an affine Lie algebra and classical integrable hierarchies associated with that Lie algebra; 6) construct hypergeometric solutions of cyclotomic KZ equations in terms of cyclotomic discriminantal arrangements; 7) use the folding relation between the Lie algebras of certain types to geometrize a type of Bethe algebra.