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Quantum magnet chains and Kashiwara crystals

October 14, 2022 @ 11:00 am - 12:00 pm

Speaker: Leonid Rybnikov, Harvard CMSA/National Research University Higher School of Economics

Title: Quantum magnet chains and Kashiwara crystals

Abstract: Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain. Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of this space (called cactus group) on the eigenlines can be described very explicitly in purely combinatorial terms of Kashiwara crystals — i.e. combinatorial objects modeling the tensor category of finite-dimensional representations of a semisimple Lie algebra g. More specifically, this monodromy action is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals. This is joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes.


October 14, 2022
11:00 am - 12:00 pm
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CMSA Room G10
CMSA, 20 Garden Street
Cambridge, MA 02138 United States
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