The proposed research is directed to developing relations between commutative algebras appearing in several fields: the algebra of Hamiltonians of a quantum integrable system, the algebra of functions on an intersection of Schubert varieties in a flag variety, the quantum cohomology algebra of a flag variety, the chiral ring of an N = 2 supersymmetric gauge theory, the local algebra of a critical point in singularity theory. All those algebras are related to the ring of functions on the critical set of the master function (or superpotential) associated with a given situation. Developing these relations will be important and useful for each of the fields and will enrich their interaction.
Multidimensional 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, 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 proposal is to develop this analysis and geometry with applications to the above theories. That will lead to better understanding of interrelations of those theories as well as to establishing new connections among them. The PI has incorporated results of his prior research into graduate courses and expects to do the same again. His graduate students and students from neighboring universities will also be involved in the research.
