Critical Points of Functions, Multidimensional Hypergeometric Integrals, and Quantum Integrable Systems

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The hypergeometric function was introduced and studied in the 18th century by Leonhard Euler. Modern versions of that function appear in different mathematical and physical theories, including representation theory, algebraic geometry, gauge theory, and statistical mechanics, and are considered from different points of view in these various research domains. The goal of this project is to develop a unified analysis and geometry of modern multidimensional hypergeometric functions with applications to these different theories. The work will lead to better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them.Multidimensional (q-)hypergeometric integrals and their semiclassical limits, Bethe eigenfunctions, and eigenvectors appear as solutions to differential and difference equations in quantum integrable systems, representation theory, algebraic geometry, gauge theory, and statistical mechanics. The equations and solutions have rich mathematical structures. The multidimensional hypergeometric integrals provide a way to transform the objects and structures of those theories to objects and structures of geometry and analysis of master functions and weight functions associated with the integrals. The goal of the project is to develop this analysis and geometry with applications to the above theories. The project involves study of representations of quantum groups, algebras of Hamiltonians of quantum integrable systems, quantum cohomology and associated quantum differential equations, Frobenius structures, the Bethe ansatz method, theory of arrangements of hyperplanes, and singularity theory of critical points of functions. The research plan is as follows: 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. Identify the elliptic Bethe algebra with the algebra of functions on the critical set of the corresponding elliptic master function; develop relations with an elliptic Schubert calculus. Identify the XXX Bethe algebra with the algebra of functions on the intersection of suitably deformed Schubert varieties. 3) Find a potential for a KZ-type connection and a Frobenius-like structure on the base of the KZ-type connection. 4) 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. Develop a correspondence between populations of critical points of master functions associated with a tensor product of irreducible modules over a simple Lie algebra and the decomposition of the tensor product into irreducibles. 5) In terms of discriminantal arrangements develop a geometric realization of the BGG resolution of an irreducible module over a simple Lie algebra.