Point vortex dynamics on K\"ahler twistor spaces

TL;DR

This paper develops tools to analyze point vortex dynamics on specific Kähler twistor spaces, providing explicit formulas and Hamiltonian equations for these complex geometric structures.

Contribution

It introduces explicit Green's function and Hamiltonian formulations for vortex dynamics on $\,\mathbb{CP}^n$ and the flag manifold, advancing understanding of vortex behavior on these spaces.

Findings

Explicit Green's function for $\,\mathbb{CP}^n$ derived

Hamiltonian equations for vortex dynamics established

Momentum map on the flag manifold determined

Abstract

In this paper, we provide tools to study the dynamics of point vortex dynamics on $\mathbb{CP}^n$ and the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$. These are the only K\"ahler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on $\mathbb{CP}^n$ which enables us to determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on $\mathbb{CP}^n$. Moreover, we determine the momentum map $\mu:\mathbb{F}_{1,2}(\mathbb{C}^3)\to \mathfrak{su}^*(3)$ on the flag manifold.