Spin Chains and Flag Manifold Sigma Models

The study of the spectrum of the Laplace-Beltrami operator on Riemannian manifolds is a classical and profound problem in differential geometry, which from a physical perspective corresponds to the question of a quantum one-dimensional sigma model on a given manifold. In general, this problem is quite difficult, and the list of manifolds for which the spectrum is known exactly is very short. Therefore, the development of new methods for investigating this spectral problem is highly relevant. This talk will consider one such method, based on the geometric quantization of coadjoint orbits, which in some cases allows one to reformulate the spectral problem for the Laplace-Beltrami operator in terms of the spectrum of the Hamiltonian of a certain spin chain. As a key example, the derivation of the Laplace operator spectrum on the sphere by diagonalizing the Hamiltonian of the SU(2) XXX spin chain with two sites in oscillator variables will be considered. A generalization of this construction to flag manifolds will also be described.