Mechanics on flag manifolds

TL;DR

This paper explores the relationship between SU(n) spin chains and sigma models on flag manifolds, calculating spectra and geodesics for specific metrics on complex projective and flag manifolds.

Contribution

It establishes a connection between spin chains and sigma models on flag manifolds and computes spectral properties and geodesics for certain metrics.

Findings

Calculated the spectrum of the Laplace-Beltrami operator.

Determined geodesics for specific metrics on complex projective and flag manifolds.

Linked spin chain models to geometric structures on flag manifolds.

Abstract

We study the connection between $\mathrm{SU}(n)$ 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 $\mathbb{CP}^1$ and $\mathcal{F}_3$, which is a manifold of complete flags in $\mathbb{C}^3$.