Bethe ansatz, Hecke algebras, and asymptotics

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The aim of the project is to study algebras of commuting Hamiltonians for quantum integrable models, called Bethe algebras, and their relations in representation theory and algebraic geometry. The main points of interest are Bethe algebras for integrable models associated with the Lie algebra gl_N, such as the Gaudin model and the XXX-type model. Recent developments show a close connection of those Bethe algebras to algebras of functions on the Calogero-Moser space and intersections of Schubert cycles in Grassmannians. One of the intended goals is to understand deeper this important connection and to expand it to other algebras of geometric nature. Another goal is to establish direct links of the Bethe algebras in question with Cherednik algebras and double affine Hecke algebras. The existence of such links is very plausible because the latter algebras are closely related to the same geometric objects as Bethe algebras. One more direction of the project is to employ ideas of quasiclassical quantization to obtain asymptotics of large eigenvalues and the corresponding eigenvectors of Bethe algebras. Hopefully, those asymptotics can be formulated in terms of solutions of classical integrable systems.The main question in the theory of quantum integrable systems is to find joint eigenvalues and eigenvectors of commuting integrals of motion. The Bethe ansatz originated as a set of clever technical tricks to perform this task. Eventually, it has developed into a variety of tools connecting this problem to many areas such as separation of variables for PDEs, Fuchsian differential equations, difference equations, Schubert calculus. This interaction turned out to be very extremely fruitful and yielded many important results for all the subjects involved. The project will contribute to further development of this area.