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Résumé |
I will describe an exact relation between 1D sigma models on homogeneous spaces
and Gaudin spin models, which in the physics setup originates from Haldane's
mapping between spin chains and sigma models. As the main example, I will
consider the sigma model with target space U(3)/U(1)^3 (the flag manifold) and
the three-site Gaudin model, respectively. The goal of the talk is to describe
an explicit solution to both the classical and quantum problems for the sigma
model: namely, finding the geodesics and calculating the spectrum of the Laplace
operator on the flag manifold equipped with an arbitrary invariant metric. This
method is also applicable to homogeneous spaces of higher-rank groups, where the
integrable case corresponds to the metrics of Manakov–Mishchenko–Fomenko
discovered in the 1970s in the study of higher-dimensional integrable
generalizations of the Euler top. |
