Mechanics on Flag Manifolds

We study the connection between $${\text{SU}}(n)$$ spin chains and one-dimensional sigma models on flag manifolds. Using this connection, we calculate the spectrum of the Laplace–Beltrami operator and geodesics for a particular class of metrics on $$\mathbb{C}{{\mathbb{P}}^{1}}$$ and $${{\mathcal{F}}_{3}}$$ , which is a manifold of complete flags in $${{\mathbb{C}}^{3}}$$ .