# Point vortex dynamics on Kähler twistor spaces

Sonja Hohloch
Sonja Hohloch
Department of Mathematics
University of Antwerp
Middelheimlaan 1
2020 Antwerp, Belgium<EMAIL_ADDRESS>and  Guner Muarem
Guner Muarem
Department of Mathematics
University of Antwerp
Middelheimlaan 1
2020 Antwerp, Belgium<EMAIL_ADDRESS>
###### 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 Kähler 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.

###### Key words and phrases:

MSC 2020: 37J37, 37J39, 53D20, 70H06; Coadjoint orbits, Green’s function,
Momentum map, Point vortex dynamics, Flag manifold.

###### Contents

  1. 1 Introduction
    1. 1.1 Point vortex dynamics
    2. 1.2 Relation to twistor spaces
    3. 1.3 Main results
    4. 1.4 Organisation of the paper
  2. 2 Preliminaries
    1. 2.1 Notions and conventions from group actions and Lie theory
    2. 2.2 Exponential of a matrix
    3. 2.3 Symplectic manifolds, Hamiltonian dynamics, and momentum maps
    4. 2.4 Weyl group and coadjoint orbits of $\mathsf{SU}(n)$
  3. 3 Geometric structures of coadjoint orbits of $\mathsf{SU}(3)$
    1. 3.1 Coadjoint orbits characterized by eigenvalues
    2. 3.2 Coadjoint orbits seen as flag manifolds
    3. 3.3 Examples of coadjoint orbits of $\mathsf{SU}(4)$
    4. 3.4 Bruhat decomposition and induced coordinates
    5. 3.5 The Kähler structure of coadjoint orbits
    6. 3.6 Different symplectic structures
  4. 4 The point vortex momentum map on $\mathbb{CP}^{2}$ and $\mathbb{F}_{1,2}(\mathbb{C}^{3})$
    1. 4.1 The momentum map for vortex dynamics
    2. 4.2 The momentum map of the degenerate orbit $\mathcal{O}_{d}^{\mathsf{SU}(3)}\simeq\mathbb{CP}^{2}$
    3. 4.3 The momentum map of the generic orbit $\mathcal{O}^{\mathsf{SU}(3)}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})$
  5. 5 Green’s function and the vortex Hamiltonian for $\mathbb{CP}^{n}$
    1. 5.1 The Laplace-Beltrami operator
    2. 5.2 The Hamiltonian for point vortex dynamics
    3. 5.3 Green’s function and the Hamiltonian on the coadjoint orbit $\mathbb{CP}^{n}$
    4. 5.4 The Hamiltonian on the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$

##  1\. Introduction

###  1.1. Point vortex dynamics

The problem of dynamics of interacting point vortices goes back to the work of
Helmoltz [Hel67] in the 19th century and can be formulated intuitively in its
simplest form as follows. Consider $N$ points $z_{1},\dots,z_{N}$ (which we
shall refer to as ‘vortices’) in the plane $\mathbb{C}\simeq\mathbb{R}^{2}$
with coordinates $z_{k}=x_{k}+iy_{k}$. Let
$\Gamma_{1},\dots,\Gamma_{N}\in\mathbb{R}^{\neq 0}$ be real, non-zero numbers
simulating the ‘vortex strength’ of each point. The equations determining this
dynamical system are given by the $N$ differential equations

$\dot{\overline{z}}_{j}=\frac{1}{2\pi
i}\sum_{k=1}^{N}\frac{\Gamma_{k}}{z_{j}-z_{k}}\qquad\mbox{for }1\leq j\leq N.$

The signs of the vortex strengths determine the sense of rotation of each
vertex. This system is in fact an example of a Hamiltonian system. Endow
$\mathbb{C}^{N}$ with the symplectic form
$\Omega:=\sum_{k=1}^{N}\Gamma_{k}\tau_{k}^{*}\omega_{st}$ where $\omega_{st}$
is the standard symplectic form on $\mathbb{C}$ and
$\tau_{k}:\mathbb{C}^{N}\to\mathbb{C}$ the projection on the $k$th component.
Denote by $r(z_{j},z_{k}):=|z_{j}-z_{k}|$ the Euclidean distance between
points $z_{j},z_{k}\in\mathbb{C}\simeq\mathbb{R}^{2}$ and set

$\operatorname{\mathsf{Diag}}_{N}(\mathbb{C}):=\\{(z_{1},\dots,z_{N})\in\mathbb{C}^{N}\mid
z_{j}=z_{k}\mbox{ for some }1\leq j,k\leq N\mbox{ with }j\neq k\\}.$

Abbreviate $z:=(z_{1},\dots,z_{N})$ and consider the Hamiltonian function
given by

$\displaystyle
H:\mathbb{C}^{N}\setminus\operatorname{\mathsf{Diag}}_{N}(\mathbb{C})\to\mathbb{R},\quad
H(z):=-\frac{1}{4\pi}\sum_{k\neq j}\Gamma_{j}\Gamma_{k}\log(r(z_{j},z_{k})).$

Identifying $z_{k}=x_{k}+iy_{k}\simeq(x_{k},y_{k})$, the Hamiltonian equations
are then given by

$\displaystyle\begin{cases}\dot{x}_{j}=-\frac{1}{\Gamma_{j}}\partial_{y_{j}}H(x_{1},\dots,x_{N},y_{1},\dots,y_{N}),\\\
\dot{y}_{j}=\frac{1}{\Gamma_{j}}\partial_{x_{j}}H(x_{1},\dots,x_{N},y_{1},\dots,y_{N}),\end{cases}\qquad\mbox{for
}1\leq j\leq N.$

Then one obtains the following three constants of motion (see for instance
Galajinsky [Gal22]) which reflect the invariance under translation and
rotation of the system:

$\displaystyle p_{x}(z):=\sum_{j=1}^{N}\Gamma_{j}\Re(z_{j}),\qquad
p_{y}(z):=\sum_{j=1}^{N}\Gamma_{j}\Im(z_{j}),\qquad
m(z):=\frac{1}{2}\sum_{j=1}^{N}\Gamma_{j}|z_{j}|^{2}$

where $\Re$ denotes the real part and $\Im$ the imaginary part of a complex
number. More generally, Lin [Lin41] showed the following to be a fitting model
for point vortex dynamics: Let $(M,\omega)$ be a symplectic manifold and
consider the space
$\mathcal{M}:=\Pi_{k=1}^{N}M\setminus\operatorname{\mathsf{Diag}}_{N}(M)$ as
phase space of $N$ moving vortices $z_{1},\dots,z_{N}$ with vortex strengths
$\Gamma_{k}\in\mathbb{R}^{\neq 0}$ for $1\leq k\leq N$. Let $G$ be the
fundamental solution (also called Green’s function) of the Laplace-Beltrami
operator and set

$\displaystyle R:M\to\mathbb{R},\qquad R(z):=\lim_{s\to
z}\left(G(s,z)-\frac{1}{2\pi}\log r(s,z)\right)$

which is often referred to as Robin function. The Hamiltonian of the system is
then given by

$\displaystyle H:\mathcal{M}\to\mathbb{R},\quad
H(z_{1},\dots,z_{N}):=\sum_{1\leq j<k\leq N}
\Gamma_{i}\Gamma_{j}G(z_{j},z_{k})+\sum_{k=1}^{N}\Gamma_{k}^{2}R_{g}(z_{k}).$

Green’s function describes the interaction between pairs of distinct vortices
and $R$ describes self-interactions of the vortices. On homogeneous manifolds,
$R$ is often neglected due to symmetry reasons.

The equations of the point vortex problem are exactly the Euler equation
arising in the discretization of fluid equations in mathematical modeling
problems, see Aref [Are07], and Angrand [ADG85] for the corresponding
numerics.

During the past years, quite some work has been done on generalizing this
approach to other symplectic manifolds, for example:

  1. (1)

On the $2$-spheres $\mathbb{S}^{2}$ with $\mathsf{SO}(3)$-invariant symplectic
form and Hamiltonian vector field, see Crowdy [Cro06], Laurent-Polz $\&$
Montaldi $\&$ Roberts [LPMR11], Lim $\&$ Montaldi $\&$ Roberts [LMR01].

  2. (2)

There has also been some research done on the cylinder concerning periodic
motion, see Montaldi $\&$ Souliere $\&$ Tokieda [MST03], Dritschel $\&$ Boatto
[DB15].

  3. (3)

Point vortices on the cylinder, see Montaldi & Souliere & Tokieda [MST03].

  4. (4)

Point vortices on the hyperbolic plane, see Montaldi $\&$ Nava-Gaxiola
[MNG14].

  5. (5)

Point vortices on $\mathbb{CP}^{2}$ with underlying symmetry group
$\mathsf{SU}(3)$, see Montaldi $\&$ Shaddad [MS19a, MS19b].

A natural question is if the examples from above can be generalised to higher
dimension. This naturally yields larger and more complicated symmetry groups.
For example, in the case of the 2-sphere, the symmetry group is
$G=\mathsf{SO}(3)$. But since the spheres $\mathbb{S}^{2n}$ for $n>1$ do not
admit a symplectic structure, generalizing straightforward to higher
dimensions with $\mathsf{SO}(m)$-symmetry does not necessarily make sense.
Nevertheless, since $\mathbb{CP}^{1}\cong\mathbb{S}^{2}$ on can intuitively
think of $\mathbb{CP}^{n}$ as the ‘best symplectic approximation’ of the
$(n+1)$-sphere, albeit with underlying higher dimensional symmetry group
$\mathsf{SU}(n)$.

###  1.2. Relation to twistor spaces

In the late seventies, Atiyah [AHS78] introduced the twistor theory for a
4-dimensional Riemannian manifold, relating it to 3-dimensional complex
analysis. A few years later, in a paper by Hitchin [Hit81], the question arose
which complex manifold could be obtained by using Atiyah’s twistor
construction on compact $4$-manifolds. More precisely, the question was: which
4-manifolds have a twistor space which is Kähler?

Surprisingly, there are not many, namely only the $4$-sphere $\mathbb{S}^{4}$
and the projective plane $\mathbb{CP}^{2}$ have Kähler twistor spaces. More
specifically, the twistor space $\mathcal{T}(\mathbb{S}^{4})$ is the complex
projective space $\mathbb{CP}^{3}$ and $\mathcal{T}(\mathbb{CP}^{2})$ is the
6-dimensional flag manifold (or Wallach space)
$\mathbb{F}_{1,2}(\mathbb{C}^{3})=\mathbb{W}^{6}=\mathsf{SU}(3)/\mathbb{T}^{2}$.
In this paper, we will be in particular interested in the spaces
$\mathbb{CP}^{n}$ and the 6-dimensional flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$ in the context of point vortex dynamics.

###  1.3. Main results

One of the goals of this paper is to obtain an explicit expression for the
Hamiltonian of the point vortex problem on certain coadjoint orbits in order
to write down explicitly the equations of motion, look for conserved
quantities, and analyse the underlying algebraic structure.

We are interested in symplectic manifolds with canonical
$\mathsf{SU}(n)$-symmetry obtained by the coadjoint action of $\mathsf{SU(n)}$
on its dual Lie algebra $\mathfrak{su}(n)^{*}$. Specifically, we will focus on
the case $n=3$. There exist exactly two coadjoint orbits, namely the six
dimensional ‘generic’ orbit

$\mathcal{O}^{\mathsf{SU}(3)}=\frac{\mathsf{SU}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)}\cong\mathbb{F}_{1,2}(\mathbb{C}^{3})$

and the four dimension ‘degenerate’ orbit (the meaning of this will become
clear later on)

$\mathcal{O}_{d}^{\mathsf{SU}(3)}=\frac{\mathsf{SU}(3)}{\mathsf{SU}(2)\times\mathsf{SU}(1)}\cong\mathbb{CP}^{2}$

where the index $d$ refers to degenerate.

The first main question that we address in this paper arose from the following
context: Montaldi $\&$ Shaddad [MS19a, MS19b] studied the point vortex
dynamics only on the degenerate orbit $\mathcal{O}_{d}^{\mathsf{SU}(3)}$ but
not on the generic one. Thus a natural question is to investigate the point
vortex problem on the generic orbit $\mathcal{O}^{\mathsf{SU}(3)}$. Before we
start with the associated momentum map we need some notation. Let $B$ be the
subgroup of upper triangular matrices of $\mathsf{SL}(3,\mathbb{C})$. Then we
may identify
$\mathbb{F}_{1,2}(\mathbb{C}^{3})\simeq\mathsf{SL}(3,\mathbb{C})/B$ (see Lemma
3.7 for details) of which the elements are of the form

$\begin{pmatrix}1&0&0\\\ z_{1}&1&0\\\
z_{2}&z_{3}&1\end{pmatrix}=:Z\in\mathsf{SL}(3,\mathbb{C})/B$

with $z_{1},z_{2},z_{3}\in\mathbb{C}$. Define the functions
$K_{1},K_{2}:\mathbb{F}_{1,2}(\mathbb{C}^{3})\to\mathbb{R}$ by

$K_{1}(Z):=1+|z_{1}|^{2}+|z_{2}|^{2}\quad\mbox{and}\quad
K_{2}(Z)=1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}.$

######  Theorem 1.1.

The momentum map of the left action of $\mathsf{SU}(3)$ on the generic
coadjoint orbit
$\mathcal{O}^{\mathsf{SU}(3)}=\mathbb{F}_{1,2}(\mathbb{C}^{3})\simeq\mathsf{SL}(3,\mathbb{C})/B$
is explicitly given by

$\displaystyle\mu:\mathsf{SL}(3,\mathbb{C})/B\to\mathfrak{su}(3)^{*},\qquad\begin{pmatrix}1&0&0\\\
z_{1}&1&0\\\ z_{2}&z_{3}&1\end{pmatrix}\mapsto(\mu_{ij})_{1\leq i,j\leq 3}$

where $(\mu_{ij})_{1\leq i,j\leq 3}$ is the traceless, anti-Hermitian matrix
with entries

$\displaystyle\mu_{11}$
$\displaystyle=\frac{1}{3}\left(\frac{x_{3}^{2}+y_{3}^{2}+2}{K_{2}}-\frac{x_{2}^{2}+y_{2}^{2}-1}{K_{1}}\right),$
$\displaystyle\mu_{22}$
$\displaystyle=\frac{1}{3}\left(-\frac{2x_{2}^{2}+2y_{2}^{2}+1}{K_{1}}-\frac{x_{3}^{2}+y_{3}^{2}-1}{K_{2}}\right),$
$\displaystyle\mu_{33}$ $\displaystyle=-(\mu_{11}+\mu_{22}),$
$\displaystyle\mu_{12}$
$\displaystyle=\frac{\left(iy_{1}-x_{1}\right)\left(x_{3}-iy_{3}\right)-iy_{2}+x_{2}}{K_{2}}-\frac{x_{1}-iy_{1}}{K_{1}},$
$\displaystyle\mu_{13}$
$\displaystyle=\frac{\left(iy_{1}-x_{1}\right)\left(x_{3}-iy_{3}\right)-iy_{2}+x_{2}}{K_{2}}-\frac{x_{1}-iy_{1}}{K_{1}},$
$\displaystyle\mu_{23}$
$\displaystyle=\frac{iy_{3}+x_{3}}{K_{2}}-\frac{\left(x_{1}+iy_{1}\right)\left(x_{2}-iy_{2}\right)}{K_{1}}.$

The remaining entries are determined by the fact that the matrix is anti-
Hermitian.

Theorem 1.1 is restated as Theorem 4.3 and proven in Section 4. We do not yet
have an explicit formula for the Hamiltonian $H$ on
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$ since this requires an explicit expression
of Green’s function on $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ which turned out to
be more involved than expected and will be treated in a future work.

The second main question that we solved in this paper was motivated by the
works of Montaldi $\&$ Shaddad [MS19a, MS19b] about the dynamics of the
generalised vortex problem on $\mathbb{CP}^{2}$ on which they worked without
an explicit expression for the Hamiltonian. In fact, it is possible to compute
the Hamiltonian of the point vortex problem for general $\mathbb{CP}^{n}$
explicitly. The answer involves first computing Green’s function on
$\mathbb{CP}^{n}$ (stated and proven as Theorem 5.5 in Section 5):

######  Theorem 1.2.

Consider $\mathbb{CP}^{n}$ with the Fubini-Study metric. Then Green’s function
is given by
$G:\left(\mathbb{CP}^{n}\times\mathbb{CP}^{n}\right)\setminus\operatorname{\mathsf{Diag}}_{2}(\mathbb{CP}^{n})\to\mathbb{R}$
with

$\displaystyle
G(\xi,\eta)=-\frac{1}{2n\cdot\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})}\left(\log(\sin(r(\xi,\eta)))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}(r(\xi,\eta))}\right)$

where
$r(\xi,\eta)=\arccos\sqrt{\frac{\langle\xi,\eta\rangle_{H}\langle\eta,\xi\rangle_{H}}{\langle\xi,\xi\rangle_{H}\langle\eta,\eta\rangle_{H}}}$
is the geodesic distance between two point in $\mathbb{CP}^{n}$ and
$\langle\cdot,\cdot\rangle_{H}$ is the Hermitian inner product.

The following theorem is stated and proven as Theorem 5.6 in Section 5.

######  Theorem 1.3.

The Hamiltonian for the $N$ point vortex dynamics on the projective space
$\mathbb{CP}^{n}$ is explicitly given by

$\displaystyle H:\
\bigl{(}\mathbb{CP}^{n}\bigr{)}^{N}\setminus\operatorname{\mathsf{Diag}}_{N}(\mathbb{CP}^{n})\
\to\ \mathbb{R},$ $\displaystyle
H(\zeta)=-\frac{1}{2(n-1)!\pi^{n}}\sum_{\alpha<\beta}^{N}\Gamma_{\alpha}\Gamma_{\beta}\left(\log(\sin(r(\zeta_{\alpha},\zeta_{\beta})))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}(r(\zeta_{\alpha},\zeta_{\beta}))}\right)$

where $\zeta=(\zeta_{1},\dots,\zeta_{N})$ and
$r(\zeta_{\alpha},\zeta_{\beta})$ is the geodesic distance on
$\mathbb{CP}^{n}$ between the two points given by

$\displaystyle
r(\zeta_{\alpha},\zeta_{\beta})=\arccos\sqrt{\frac{\langle\zeta_{\alpha},\zeta_{\beta}\rangle_{H}\langle\zeta_{\beta},\zeta_{\alpha}\rangle_{H}}{\langle\zeta_{\alpha},\zeta_{\alpha}\rangle_{H}\langle\zeta_{\beta},\zeta_{\beta}\rangle_{H}}},$

where $\langle\cdot,\cdot\rangle_{H}$ is the Hermitian inner product.

As already explained above, we do not yet have an explicit formula for the
Hamiltonian $H$ on $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ since we do not yet have
an explicit expression of Green’s function on
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$. The hope (see Section 5.4) is to make use
of the fibration

$\mathbb{S}^{2}\longrightarrow\mathbb{F}_{1,2}(\mathbb{C}^{3})\longrightarrow\mathbb{C}\mathbb{P}^{2}$

and to obtain Green’s function on $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ from
Green’s functions on $\mathbb{S}^{2}$ and $\mathbb{C}\mathbb{P}^{2}$ and thus
obtain the Hamiltonian on $\mathbb{F}_{1,2}(\mathbb{C}^{3})$. This is planned
to be carried out in a future work.

###  1.4. Organisation of the paper

We give a quick overview of this paper:

  * •

In Section 2, we recall necessary notions and results from Lie algebras,
representation theory and differential geometry.

  * •

In Section 3, we consider geometric and algebraic features and properties of
the two coadjoint orbits
$\mathcal{O}^{\operatorname{\mathsf{SU}}(3)}_{d}\simeq\mathbb{CP}^{2}$ and
$\mathcal{O}^{\operatorname{\mathsf{SU}}(3)}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})$.

  * •

In Section 4, we analyse the momentum map for various situations.

  * •

In Section 5, we study Green’s function in various settings and compute an
explicit formula for it on $\mathbb{CP}^{n}$. After that, we compute the
associated Hamiltonian function.

Acknowledgments. The authors wish to thank Marine Fontaine for helpful
discussions and useful comments. S. Hohloch was partially and G. Muarem was
fully supported by the FWO-EoS project ‘Symplectic Techniques in Differential
Geometry’ G0H4518N.

##  2\. Preliminaries

###  2.1. Notions and conventions from group actions and Lie theory

Let $G$ be a compact Lie group with Lie algebra $\mathsf{Lie}(G)=\mathfrak{g}$
and dual algebra $\mathfrak{g}^{*}$. The (left) action of a Lie group $G$ on a
manifold $M$ is denoted by the map $\Phi:G\times M\to M$ which satisfies for
all $x\in M$

  1. (1)

$\Phi(e,x)=x$,

  2. (2)

$\Phi(g,\Phi(h,x))=\Phi(gh,x)$ for all $g,h\in G$.

We usually write the action briefly as $g\cdot x$ or simply $gx$. Recall that
Lie group actions are smooth. Moreover, the isotropy subgroup or stabilizer of
a point $m$ is given by the closed subgroup $G_{m}:=\\{g\in G\mid gm=m\\}$.
The orbit under $G$ of a point $m\in M$ is given by

$\mathcal{O}_{m}:=\\{gm\in M\mid g\in G\\}\subseteq M.$

Lying in the same orbit gives rise to an equivalence relation on the manifold
$M$ via $x\sim y\Leftrightarrow gx=y$ for $x,y\in M$ and $g\in G$. The space
consisting of all these equivalence classes is called the orbit space and
denoted by $M/G$. Now consider the action of a Lie group $G$ on itself by
conjugation

$c_{g}:G\to G,\quad h\mapsto ghg^{-1}.$

Identifying the Lie algebra $\mathfrak{g}$ with the tangent space $T_{e}G$ at
the neutral element $e\in G$ and differentiating $c_{g}$ in $e$, we obtain for
all $g\in G$

$\operatorname{\mathsf{Ad}}_{g}:T_{e}G\simeq\mathfrak{g}\ \to\
T_{e}G\simeq\mathfrak{g},\quad{\operatorname{\mathsf{Ad}}_{g}X=gXg^{-1}}$

with adjoint representation

$\operatorname{\mathsf{Ad}}:G\times\mathfrak{g}\to\mathfrak{g},\quad(g,X)\mapsto\operatorname{\mathsf{Ad}}_{g}(X).$

As dual notation, we have the coadjoint representation

$\textsf{Ad}^{*}:G\times\mathfrak{g}^{*}\to\mathfrak{g}^{*},\quad(g,\alpha)\mapsto\mathsf{Ad}^{*}_{g^{-1}}\alpha=g\alpha
g^{-1}.$

For every $\mu\in\mathfrak{g}^{*}$, the set

$\mathcal{O}_{\mu}:=\\{\operatorname{\mathsf{Ad}}_{g}^{*}\mu\mid\text{for all
}g\in G\\}\subseteq\mathfrak{g}^{*}$

is the coadjoint orbit of $G$ through $\mu$.

Note that, in general, the adjoint and coadjoint representations (and thus the
resulting orbits) are not isomorphic, see for instance counterexamples given
by groups of Euclidean type (cf. Arathoon $\&$ Montaldi [AM18]).

In this paper, we are working with Lie algebras that consist of matrices. Here
the dual pairing between $\mathfrak{g}$ and $\mathfrak{g}^{*}$ is given by the
so-called Killing form $\kappa(\cdot,\cdot)$ which is a multiple of the trace
of the product of the two matrices. For example, for $\mathfrak{su}(n)$, the
Killing form is given by

$\kappa_{\mathfrak{su}(n)}(X,Y)=\operatorname{\mathsf{trace}}(XY)\quad\mbox{for
all }X,Y\in\mathfrak{su}(n).$ (2.1.1)

We denote the pairing between $\mathfrak{g}$ and $\mathfrak{g}^{*}$ by

$\langle\cdot,\cdot\rangle:\mathfrak{g}\to\mathfrak{g}^{*},\quad
X\mapsto\kappa_{X}\mbox{ where }\kappa_{X}(Y):=\kappa(X,Y)\mbox{ for all
}X,Y\in\mathfrak{g}.$

###  2.2. Exponential of a matrix

In the context of Lie groups and Lie algebras, the exponential map is defined
as

$\displaystyle\exp:\mathfrak{g}\to G,\quad X\mapsto\gamma(1)$

where $\gamma:\mathbb{R}\to G$ is the unique one-parameter subgroup of $G$ for
which the tangent vector at the identity is $X$. In the case of a matrix Lie
group, the exponential map is given by

$\displaystyle\exp(A):=\sum_{k=0}^{\infty}\frac{1}{k!}A^{k}\quad\mbox{for all
}(n\times n)\mbox{-matrices }A,$

briefly called the exponential of the matrix $A$. If
$A=\operatorname{\mathsf{Diag}}(a_{11},\dots,a_{nn})$ is a diagonal matrix we
obtain $A^{k}=\operatorname{Diag}(a^{k}_{11},\dots,a^{k}_{nn})$ and therefore

$\displaystyle\exp(A)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}{\operatorname{\mathsf{Diag}}}(a^{k}_{11},\dots,a^{k}_{nn})=\operatorname{\mathsf{Diag}}\left(\sum_{k=0}^{\infty}\frac{1}{k!}a_{11}^{k},\dots,\sum_{k=0}^{\infty}\frac{1}{k!}a_{nn}^{k}\right)$
$\displaystyle=\operatorname{\mathsf{Diag}}\left(e^{a_{11}},\dots,e^{a_{nn}}\right).$

If $A$ is diagonalisable with $A=PDP^{-1}$, where $P$ is the matrix of
eigenvectors and $D$ is the diagonal matrix with the eigenvalues on the
diagonal, then

$\displaystyle A^{k}=(PDP^{-1})^{k}=PDP^{-1}\cdots PDP^{-1}=PD^{k}P^{-1}$

and therefore

$\displaystyle\exp(A)=P\exp(D)P^{-1}.$ (2.2.1)

###  2.3. Symplectic manifolds, Hamiltonian dynamics, and momentum maps

A symplectic manifold $(M,\omega)$ is a smooth manifold $M$ equipped with a
symplectic form $\omega$ which is a closed non-degenerate differential
$2$-form, i.e. $d\omega=0$ and whenever $\omega_{x}(u,v)=0$ for all $u\in
T_{x}M$ then $v=0$. This implies in particular that symplectic manifolds are
always even dimensional.

Given a symplectic manifold $(M,\omega)$, the map

$TM\to T^{*}M,\quad
X\to\iota_{X}\omega\quad\mbox{where}\quad\iota_{X}\omega(Y):=\omega(X,Y)\mbox{
for all }Y\in TM$

is an isomorphism, often referred to as contraction of $\omega$ by a vector
field. Symplectic manifolds are the natural geometric background for
Hamiltonian dynamics: Given a smooth function $H:M\to\mathbb{R}$, its
Hamiltonian vector field (also called symplectic gradient) is defined via
$\iota_{X_{H}}\omega=dH$. In this situation, $H$ is often referred to as
Hamiltonian function. Let $G$ be a Lie group $G$ with Lie algebra
$\mathsf{Lie}(G)=\mathfrak{g}$ and assume that the action $G\times M\to M$ is
by symplectomorphisms, i.e, for all $g\in G$, the map $M\to M$, $x\mapsto g.x$
is a symplectomorphism. Denote by $\langle\cdot,\cdot\rangle$ the dual pairing
$\langle\cdot,\cdot\rangle:\mathfrak{g}^{*}\times\mathfrak{g}\to\mathbb{R}$.
Every $\xi\in\mathfrak{g}$ gives rise to a vector field $X_{\xi}$ via

$X_{\xi}(x)=\frac{d}{dt}{\bigg{|}_{t=0}}\exp(t\xi)\cdot x$

for all $x\in M$. The momentum map for this $G$-action on $(M,\omega)$ is a
map $\mu:M\to\mathfrak{g}^{*}$ such that

$d(\langle\mu,\xi\rangle)=\iota_{X_{\xi}}\omega$

for all $\xi\in\mathfrak{g}$ where

$\langle\mu,\xi\rangle:M\to\mathbb{R},\quad x\mapsto\langle\mu(x),\xi\rangle.$

###  2.4. Weyl group and coadjoint orbits of $\mathsf{SU}(n)$

In this paper, we are in particular interested in the coadjoint orbits of the
Lie group $\mathsf{SU}(3)$ since they are can be chosen to be the setting for
vortex dynamics on the projective plane $\mathbb{CP}^{2}$ and the flag
manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$.

Let us start with fixing notation and recalling some properties of this Lie
group and its Lie algebra. The general linear group is defined as

$\displaystyle\mathsf{GL}(n,\mathbb{C})=\\{A\in\operatorname{Mat}_{n}(\mathbb{C})\mid\det
A\neq 0\\}.$

The maximal compact simply connected Lie subgroup of
$\mathsf{GL}(n,\mathbb{C})$ is given by

$\mathsf{SU}(n)=\\{U\in\operatorname{Mat}_{n}(\mathbb{C})\mid
U\overline{U}^{T}=1,\ \det U=1\\}.$

We have $\dim_{\mathbb{R}}\mathsf{SU}(n)=n^{2}-1$. The Lie algebra
$\mathfrak{su}(n)$ of $\mathsf{SU}(n)$ can be identified with

$\\{U\in\operatorname{Mat}_{n}(\mathbb{C})\mid\overline{U}^{T}=-U,\
\operatorname{\mathsf{trace}}{U}=0\\},$

meaning all $(n\times n)$-matrices which are skew-Hermitian matrices with
trace zero. Our convention for the Lie bracket is $[A,B]=AB-BA$ for all
$A,B\in\mathfrak{su}(n)$. We say that $A,B\in\mathfrak{su}(n)$ commute if
$[A,B]=0$.

In what follows, we will often work with the following basis of
$\mathfrak{su}(3)$:

######  Notation 2.1.

The following eight traceless traceless $(3\times 3)$-matrices are known as
the Gell-Mann matrices.

$\displaystyle\tilde{\lambda}_{1}$ $\displaystyle={\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&0\end{pmatrix}}$ $\displaystyle\tilde{\lambda}_{2}$
$\displaystyle={\begin{pmatrix}0&-i&0\\\ i&0&0\\\ 0&0&0\end{pmatrix}}$
$\displaystyle\tilde{\lambda}_{3}$ $\displaystyle={\begin{pmatrix}1&0&0\\\
0&-1&0\\\ 0&0&0\end{pmatrix}}$ $\displaystyle\tilde{\lambda}_{4}$
$\displaystyle={\begin{pmatrix}0&0&1\\\ 0&0&0\\\ 1&0&0\end{pmatrix}}$
$\displaystyle\tilde{\lambda}_{5}$ $\displaystyle={\begin{pmatrix}0&0&-i\\\
0&0&0\\\ i&0&0\end{pmatrix}}$ $\displaystyle\tilde{\lambda}_{6}$
$\displaystyle={\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}}$
$\displaystyle\tilde{\lambda}_{7}$ $\displaystyle={\begin{pmatrix}0&0&0\\\
0&0&-i\\\ 0&i&0\end{pmatrix}}$ $\displaystyle\tilde{\lambda}_{8}$
$\displaystyle={\frac{1}{\sqrt{3}}}{\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}}.$

We have $[\tilde{\lambda}_{3},\tilde{\lambda}_{8}]=0$ and no other of these
matrices commute with both $\tilde{\lambda}_{3}$ and $\tilde{\lambda}_{8}$.
The set

$\left\\{\left.\lambda_{k}:=\frac{i}{2}\tilde{\lambda}_{k}\ \right|\
k=1,\dots,8\right\\}$

forms a (rescaled) basis for the Lie algebra $\mathfrak{su}(3)$, often called
Gell-Mann basis.

######  Remark 2.2.

The fact that only $\lambda_{3}$ and $\lambda_{8}$ commute (see Notation 2.1)
implies that $\mathsf{SU}(3)$ has rank two so that there are two Casimirs
denoted by $C_{1},C_{2}\in\mathfrak{su}(3)$, i.e.,
$[C_{1},\lambda_{k}]=[C_{2},\lambda_{k}]=0$ for all $k=1,\dots,8$. Explicitly,
we have $C_{1}=\sum_{k=1}^{8}\lambda_{k}\lambda_{k}$ and
$C_{2}=8\sum_{j,k,\ell=1}^{8}d_{jk\ell}\lambda_{j}\lambda_{k}\lambda_{\ell}$
where $I_{3}$ is the $(3\times 3)$ unit matrix and $d_{jk\ell}$ the so-called
structure constants of $\mathfrak{su}(3)$.

We will now recall the so-called Weyl group. Let $E$ be a finite dimensional
vector space over $\mathbb{R}$ and let $\langle\cdot,\cdot\rangle_{E}$ be an
inner product on $E$. A roots system $\Phi\subset E$ is a finite set of non-
zero vectors, called roots such that:

  1. (1)

The set of roots $\Phi$ spans the space $E$.

  2. (2)

If $\alpha\in\Phi$ and $c\in\mathbb{R}$, then $c\alpha\in\Phi$ if and only if
$c=\pm 1$.

  3. (3)

$\sigma_{\alpha}(\beta):=\beta-2\frac{\langle\alpha,\beta\rangle_{E}}{\langle\alpha,\alpha\rangle_{E}}\alpha\in\Phi$
for all $\alpha,\beta\in\Phi$, i.e. $\Phi$ is invariant under
$\sigma_{\alpha}$ for all $\alpha\in\Phi$ which is the reflection about the
hyperplane orthogonal to $\alpha$.

  4. (4)

$(\alpha,\beta):=2\frac{\langle\alpha,\beta\rangle_{E}}{\langle\alpha,\alpha\rangle_{E}}\in\mathbb{Z}$
for all $\alpha,\beta\in\Phi$, i.e. the projection of $\beta$ onto the line
through $\alpha$ is an integer or half-integer multiple of $\alpha$.

A subset $\Phi^{+}\subset\Phi$ is called a positive root system if

  1. (1)

for all $\alpha\in\Phi$, either $\alpha\in\Phi$ or $-\alpha\in\Phi$,

  2. (2)

for all $\alpha,\beta\in\Phi^{+}$, we have $\alpha+\beta\in\Phi^{+}$.

Denote by $\mathsf{O}(E):=\\{A\in\mathsf{GL}(E)\mid\langle
Av,Aw\rangle_{E}=\langle v,w\rangle_{E},\text{ for all }v,w\in E\\}$ the
orthogonal group consisting of all elements in $E$ preserving the inner
product. The (finite) subgroup $\mathcal{W}\leq\mathsf{O}(E)$ generated by all
reflections $\sigma_{\alpha}$ with $\alpha\in\Phi$ is called the Weyl group
associated to $\Phi$. Denote the hyperplane perpendicular to $\alpha\in\Phi$
by $\Pi_{\alpha}$. The closure of a connected component of
$E\setminus\\{\Pi_{\alpha}\mid\alpha\in\Phi\\}$ is called a Weyl chamber. We
define the positive Weyl chamber (with respect to a fixed choice of
$\Phi^{+}$) as the closed set

$\mathcal{C}=\\{x\in E\mid\langle x,\alpha\rangle_{E}\geq 0\text{ for all
$\alpha\in\Phi^{+}$}\\}.$

Given a positive root system, there is only one positive Weyl chamber. Let $G$
be a Lie group and consider a maximal torus $H\subset G$ (i.e. a compact,
connected, abelian Lie subgroup of $G$ which is maximal with respect to these
properties) and the Cartan algebra $Lie(H):=\mathfrak{h}\subset\mathfrak{g}$
with dual $\mathfrak{h}^{*}$. Note that a maximal torus is unique up to
conjugation. In the case of $\mathsf{SU}(n)$ the situation is as follows:

######  Example 2.3.

The maximal torus $T$ of $\operatorname{\mathsf{SU}}(n)$ is given by the
diagonal matrices
$\operatorname{\mathsf{Diag}}(e^{i\theta_{1}},\dots,e^{i\theta_{n}})$ such
that $\prod_{j=1}^{n}e^{i\theta_{j}}=1$. The Lie algebra
$\mathsf{Lie}(T)=:\mathfrak{t}$ (i.e. the Cartan algebra) is then given by the
space of traceless diagonal matrices
$\mathfrak{t}=\left\\{\operatorname{\mathsf{Diag}}(\theta_{1},\dots,\theta_{n})\left|\
\sum_{j=1}^{n}\theta_{j}=0\right.\right\\}$. Thus, the interior
$\mathfrak{t}_{+}^{0}$ of the positive Weyl chamber $\mathfrak{t}_{+}$ is
given by

$\mathfrak{t}_{+}=\left\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\left|\
x_{1}>x_{2}>\cdots>x_{n}\text{ and }\sum x_{i}=0\right.\right\\}$

and the closure $\overline{\mathfrak{t}}_{+}=\mathfrak{t}_{+}$ is given by
replacing $>$ by $\geq$ in the above set. Here the Weyl group is the symmetric
group $\mathsf{Sym}(3)$ generated by the positive roots $\alpha$, $\beta$,
$\alpha+\beta$ sketched in Figure 1. It permutes in fact all roots.

$\alpha$$\beta$$\alpha+\beta$$-\alpha$$-\alpha-\beta$$-\beta$ Figure 1. Root
diagram of $\mathfrak{su}(3)$ where the positive roots are given by
$\alpha=(1,0)$, $\beta=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ and
$\alpha+\beta=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. The simple roots
are given by $\alpha$ and $\beta$. The blue shaded area is the positive Weyl
chamber.

The orbits of the Weyl group are described by Bott’s theorem:

######  Theorem 2.4 (Bott).

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and Cartan subalgebra
$\mathfrak{h}$ and dual $\mathfrak{h}^{*}$. Let $\mathcal{O}_{\mu}$ be a
coadjoint orbit of $G$. Then $\mathcal{O}_{\mu}\cap\mathfrak{h}^{*}$ is an
orbit of the Weyl group.

This implies that each coadjoint orbit $\mathcal{O}$ of $G$ is uniquely
defined by a starting point $\mu_{0}\in\mathfrak{h}^{*}$ in the (closed)
positive Weyl chamber (for more details, see for example Bernatska $\&$ Holod
et al. [BH08]).

######  Definition 2.5.

Denote the interior of the positive Weyl chamber $W$ by $W^{\circ}$ and its
boundary by $\partial W$. The coadjoint orbit of a point $\mu_{0}\in
W^{\circ}$ is said to be a generic orbit. The coadjoint orbit of a point
$\mu_{0}\in\partial W$ is called a degenerate orbit.

######  Example 2.6.

The group $\operatorname{\mathsf{SU}}(3)$ has exactly two coadjoint orbits, a
generic one of dimension six and a degenerate one of dimension four. The
generic orbit is denoted by $\mathcal{O}^{\mathsf{SU}(3)}$ and can be
identified with

$\mathcal{O}^{\mathsf{SU}(3)}=\frac{\mathsf{SU}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)}.$

The degenerate orbit is denoted by $\mathcal{O}^{\mathsf{SU}(3)}_{d}$ and can
be identified with

$\mathcal{O}_{d}^{\mathsf{SU}(3)}=\frac{\mathsf{SU}(3)}{\mathsf{SU}(2)\times\mathsf{SU}(1)}\cong\mathbb{CP}^{2}.$

##  3\. Geometric structures of coadjoint orbits of $\mathsf{SU}(3)$

In this section, we characterize coadjoint orbits of $\mathsf{SU}(3)$ by their
algebraic and geometric properties and, eventually, describe their Kähler
structure.

###  3.1. Coadjoint orbits characterized by eigenvalues

Let $(X,\\{\cdot,\cdot\\})$ be a Poisson manifold. A maximal connected
submanifold $Y\subset X$ for which the Poisson structure descends to a
symplectic structure is called a symplectic leaf. Moreover, the Poisson
manifold is foliated by its symplectic leaves. Let $\mathfrak{g}$ be a Lie
algebra with dual $\mathfrak{g}^{*}$. Then there is a canonical Poisson
structure on $\mathfrak{g}^{*}$ called the Lie-Poisson structure. In this
case, the symplectic leaves are the coadjoint orbits. A smooth function
$C:\mathfrak{g}\to\mathbb{R}$ is called a Casimir function if $C$ is constant
on each coadjoint orbit, or equivalently, if $C$ is invariant under the
coadjoint action of $G$ on $\mathfrak{g}^{*}$. Recall that level sets of
Casimir functions $C:\mathfrak{g}^{*}\to\mathbb{R}$ are symplectic manifolds
(see for instance Arnaudon $\&$ De Castro $\&$ Holm [ADCH18]) and that
coadjoint orbits lie on level sets of Casimir functions (since they are
invariant under the coadjoint action).

In the case of $\mathfrak{su}(n)$, the Casimir functions are given by

$C_{j}:\mathfrak{su}^{*}(n)\to\mathbb{R},\quad
A\mapsto\operatorname{\mathsf{trace}}(A^{j})$

for $j=1,\dots,n-1$ where $A^{j}:=A\circ\dots\circ A$ is the $j$-fold
composition. For $n=3$, we may therefore study the dynamics on the
intersection of the Casimir level sets $C_{1}=c_{1}$ and $C_{2}=c_{2}$ for
some constants $c_{1}$ and $c_{2}$. The set
$\\{C_{1}=c_{1}\\}\cap\\{C_{2}=c_{2}\\}$ can be identified with the space

$\mathcal{M}:=\\{A\in\operatorname{Mat}_{3}(\mathbb{C})\mid
A=\overline{A}^{T},\ \operatorname{\mathsf{trace}}(A)=0,\
\operatorname{\mathsf{trace}}(A^{2})=c_{1},\
\operatorname{\mathsf{trace}}(A^{3})=c_{2}\\}$

which can be seen as phase space of a
$\operatorname{\mathsf{SU}}(3)$-invariant dynamical system. In the generic
case, we have $\dim(\mathcal{M})=6$ and, in the degenerate case,
$\dim(\mathcal{M})=4$ which corresponds to the dimensions of two coadjoint
orbits of $\mathsf{SU}(3)$, see Example 2.6.

######  Lemma 3.1.

The phase space $\mathcal{M}$ is isomorphic as vector space to the space

$\\{A\in\operatorname{Mat}_{3}(\mathbb{C})\mid A=\overline{A}^{T},A\text{ has
eigenvalues }\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\text{ with
}\lambda_{1}+\lambda_{2}+\lambda_{3}=0\\}.$

###### Proof.

The characteristic polynomial $\chi_{A}$ for a traceless Hermitian matrix $A$
can be written non-factorized and factorized:

$\chi_{A}=\det(A-\lambda
I_{3})=-\lambda^{3}-\frac{1}{2}\operatorname{\mathsf{trace}}(A^{2})\lambda+\frac{1}{3}\operatorname{\mathsf{trace}}(A^{3})=(\lambda_{1}-\lambda)(\lambda_{2}-\lambda)(\lambda_{3}-\lambda).$

In the non-factorized version, two coefficients are expressed by the trace
and, in the factorized version, the eigenvalues of the matrix appear.
Comparing the coefficients in the expression above with the definition of
$\mathcal{M}$ and the claim in the statement yields the bijection. ∎

Let
$\Lambda:=\operatorname{\mathsf{Diag}}(\lambda_{1},\lambda_{2},\lambda_{3})$
with $(\lambda_{1},\lambda_{2},\lambda_{3})\in\mathbb{R}^{3}$ and set
$\mathcal{O}_{\Lambda}=\\{A\Lambda A^{-1}\mid
A\in\operatorname{\mathsf{SU}}(3)\\}$. The spectral theorem for Hermitian
matrices states that the eigenvalues of a Hermitian matrix are real and that
the eigenvectors corresponding to these eigenvalues are orthogonal. This
allows to deduce the following bijective correspondence:

$\displaystyle\begin{Bmatrix}\text{coadjoint orbits}\\\ \text{of
}\mathsf{SU}(3)\end{Bmatrix}$
$\displaystyle\longleftrightarrow\begin{Bmatrix}\mathcal{O}_{\Lambda}\text{
with }\lambda_{1}+\lambda_{2}+\lambda_{3}=0\\\ \text{and
}\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\end{Bmatrix}$

This leads to three types orbits (of which one is trivial):

  1. (i)

All three eigenvalues are distinct. Then the stabilizer is given by
$\operatorname{\mathsf{Diag}}(\alpha,\beta,\overline{\alpha\beta})$ with
$\alpha,\beta\in\mathbb{C}$ and the coadjoint orbit is

$\mathcal{O}^{\operatorname{\mathsf{SU}}(3)}=\frac{\operatorname{\mathsf{SU}}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)}=\frac{\operatorname{\mathsf{U}}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)}.$

We will see in Section 3.2 that this orbit can be identified with a six
dimensional flag manifold.

  2. (ii)

Two eigenvalues are equal. The stabilizer is given by the block diagonal
matrix $\operatorname{\mathsf{Diag}}(A,\overline{\det A})$ where
$A\in\operatorname{\mathsf{U}}(2)$. In this case the coadjoint orbit can be
identified with

$\mathcal{O}^{\operatorname{\mathsf{SU}}(3)}_{d}=\operatorname{\mathsf{SU}}(3)/\operatorname{\mathsf{U}}(2)\cong\mathbb{CP}^{2}.$

  3. (iii)

All eigenvalues are equal. In this case, the stabilizer is
$\operatorname{\mathsf{SU}}(3)$ so that the orbit is trivial (since
$\lambda_{1}+\lambda_{2}+\lambda_{3}=3\lambda=0$ implies $\lambda=0$).

######  Remark 3.2.

More generally, setting
$\Lambda:=\operatorname{\mathsf{Diag}}(\lambda_{1},\dots,\lambda_{n})$ with
$(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$ and
$\lambda_{1}\geq\cdots\geq\lambda_{n}$ and $\sum_{i=1}^{n}\lambda_{i}=0$, the
coadjoint orbits of $\operatorname{\mathsf{SU}}(n)$ are of the form
$\mathcal{O}_{\Lambda}=\\{A\Lambda A^{-1}\mid
A\in\operatorname{\mathsf{SU}}(n)\\}$.

###  3.2. Coadjoint orbits seen as flag manifolds

The four dimensional degenerate orbit space has a nice geometrical
interpretation as $\mathbb{CP}^{2}$. We will now see that there is also a nice
geometric characterisation of the generic orbit as a so-called flag manifold
(of which the degenerate orbit $\mathbb{CP}^{2}$ is a special case).

######  Definition 3.3.

Consider $\mathbb{C}^{n}$ and let $r\in\\{1,\dots,n\\}$. A flag
$f_{{k_{1}},\dots,{k_{r}};n}$ in $\mathbb{C}^{n}$ is a nested sequence of
vector subspaces $V_{k_{1}}\subsetneq\dots\subsetneq V_{k_{r}}$ in
$\mathbb{C}^{n}$ such that $\dim_{\mathbb{C}}V_{k_{j}}=k_{j}$ for all $1\leq
j\leq r$. The space of all such flags is denoted by
$\mathbb{F}_{{k_{1}},\dots,{k_{r}}}(\mathbb{C}^{n})$.

######  Remark 3.4.

$\mathbb{F}_{{k_{1}},\dots,{k_{r}}}(\mathbb{C}^{n})$ is a compact, complex and
smooth manifold and is usually referred to as the flag manifold. Note that all
flag manifolds are in fact generalisations of projective spaces. The flag
manifold $\mathbb{F}_{1}(\mathbb{C}^{n})$ is precisely
$\mathbb{C}\mathbb{P}^{n-1}$. Moreover, the flag manifold
$\mathbb{F}_{k}(\mathbb{C}^{n})$ is the space of $k$-dimensional vector
subspaces of $\mathbb{C}^{n}$, i.e., the Grassmannian.

For $n=3$, $k_{1}=1$ and $k_{2}=2$, we obtain the generic coadjoint orbit of
$\mathsf{SU}(3)$

$\displaystyle\mathcal{O}^{\operatorname{\mathsf{SU}}(3)}=\mathbb{F}_{1,2}(\mathbb{C}^{3})=\\{(L,P)\mid
L\subset P\subset\mathbb{C}^{3}\text{ with
}\dim_{\mathbb{C}}(L)=1,\dim_{\mathbb{C}}(P)=2\\}.$

This space also appears in the context of so-called Wallach manifolds
introduced by Wallach [Wal72] which we will describe now. Consider the linear
map $J:\mathbb{C}^{2n}\to\mathbb{C}^{2n}$ defined as

$J(z_{1},\dots,z_{n},z_{n+1},\dots,z_{2n})=(z_{2n},\dots,z_{n+1},-z_{n},\dots,-z_{1}).$

The group $\mathsf{Sp}(n)$ is defined as

$\mathsf{Sp}(n)=\\{A\in\mathsf{SU}(2n)\mid AJ=J\bar{A}\\}.$

Let $\mathsf{F}(4)$ be the 52-dimensional exceptional simple Lie group.
Moreover, recall that the universal cover of the orthogonal group
$\mathsf{SO}(8)$ is called the spin group and is denoted by
$\mathsf{Spin}(8)$. The Wallach manifolds $W^{6}$ of dimension six, $W^{12}$
of dimension twelve, and $W^{24}$ of dimension twenty-four are given by

$\displaystyle
W^{6}:=\frac{\operatorname{\mathsf{SU}}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)},\quad
W^{12}:=\frac{\operatorname{\mathsf{Sp}}(3)}{\operatorname{\mathsf{Sp}}(1)\times\operatorname{\mathsf{Sp}}(1)\times\operatorname{\mathsf{Sp}}(1)}\quad\text{and}\quad
W^{24}:=\frac{\mathsf{F}(4)}{\operatorname{\mathsf{Spin}}(8)}.$

These are all compact Riemannian manifolds of positive curvature. Moreover,
these manifolds can be thought of as the total space of the following
homogeneous fibrations:

$\displaystyle\mathbb{S}^{2}$ $\displaystyle\longrightarrow
W^{6}\longrightarrow\mathbb{C}\mathbb{P}^{2},$ $\displaystyle\mathbb{S}^{4}$
$\displaystyle\longrightarrow W^{12}\longrightarrow\mathbb{HP}^{2},$
$\displaystyle\mathbb{S}^{8}$ $\displaystyle\longrightarrow
W^{24}\longrightarrow\mathbb{O}\mathbb{P}^{2}.$

For more details on these fibrations, we refer the reader to Dearricott $\&$
Galaz-Garciá et al. [Dea14] and the references therein.

######  Remark 3.5.

On each flag manifold there is a natural action of the isometry group of
$\mathbb{C}\mathbb{P}^{2}$, resp. $\mathbb{HP}^{2}$, resp.
$\mathbb{O}\mathbb{P}^{2}$. When looking at these fibrations, one might ask if
it is possible to generalise the point vortex dynamics to $\mathbb{HP}^{2}$
and $\mathbb{OP}^{2}$. However, neither $\mathbb{HP}^{2}$ nor
$\mathbb{OP}^{2}$ admit a symplectic structure since $\mathbb{HP}^{2}$ is a
quaternionic Kähler manifold and these are not symplectic. Moreover, any
symplectic manifold admits compatible almost-complex structures, but
$\mathbb{OP}^{2}$ does not admit one.

###  3.3. Examples of coadjoint orbits of $\mathsf{SU}(4)$

In this article, we mainly focus on the two coadjoint orbits of
$\mathsf{SU}(3)$ given by $\mathbb{CP}^{2}$ and
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$. Now we want to have a brief glance at the
situation for $\mathsf{SU}(4)$ (see Bernatska & Holod [BH08]).

coadjoint orbit | type | dimension | name
---|---|---|---
$\operatorname{\mathsf{SU}}(4)/\operatorname{\mathsf{U}}(1)^{3}$ | generic | 12 |  $\mathbb{F}_{1,2,3}(\mathbb{C}^{4})$ (‘full flag’)
$\operatorname{\mathsf{SU}}(4)/\operatorname{\mathsf{SU}}(2)\times\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)$ | degenerate | 10 |  $\mathbb{F}_{1,2}(\mathbb{C}^{4})$ (‘partial flag’)
$\operatorname{\mathsf{SU}}(4)/S(\operatorname{\mathsf{U}}(2)\times\operatorname{\mathsf{U}}(2))$ | degenerate | 8 |  $\operatorname{\mathsf{Gr}}_{2}(\mathbb{C}^{4})$ (Grassmannian)
$\operatorname{\mathsf{SU}}(4)/\operatorname{\mathsf{SU}}(3)\times\operatorname{\mathsf{U}}(1)$ | degenerate | 6 |  $\mathbb{CP}^{3}$ (projective space)

The ‘partial flag manifold’ $\mathbb{F}_{1,2}(\mathbb{C}^{4})$ consists of
pairs $(L,P)$ where $L$ is a one-dimensional and $P$ is a two-dimensional
subspace of $\mathbb{C}^{4}$ such that $L\subset P$. There are natural
projections $\varphi_{1}(L,P)=L$ and $\varphi_{2}(L,P)=P$.

A fibre bundle over $A$ with fibre $B$ with map $\tau$ is denoted by
$\mathcal{E}(A,B,\tau)$. We have in particular that the generic orbit is a
fibre bundle over a degenerate orbit, more precisely (see Bernatska $\&$ Holod
[BH08])

$\displaystyle\frac{\operatorname{\mathsf{SU}}(4)}{\operatorname{\mathsf{U}}(1)^{3}}=\mathcal{E}(\mathbb{CP}^{3},\mathbb{F}_{1,2}(\mathbb{C}^{3}),\tau_{2})=\mathcal{E}(\operatorname{\mathsf{Gr}}_{2}(\mathbb{C}^{4}),\mathbb{CP}^{1},\tau_{2}),$

and the three degenerate spaces are in connection with each other by the
following double fibration:

${\mathbb{F}_{1,2}(\mathbb{C}^{4})}$${\mathbb{CP}^{3}}$${\operatorname{\mathsf{Gr}}_{2}(\mathbb{C}^{4})}$$\tau_{2}$$\tau_{1}$

As a general remark, the partial flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{4})$ can also be described in terms of algebraic
geometry by using the Plücker embedding. In the case of the Grassmannian this
embeddings has an interesting interpretation:

$\operatorname{\mathsf{Gr}}_{2}(\mathbb{C}^{4})\hookrightarrow\mathbb{P}\left({\bigwedge}^{2}\mathbb{C}^{4}\right)\cong\mathbb{CP}^{5}.$

Thus, the image of the Grassmannian is a quadric in $\mathbb{CP}^{5}$, often
called the Klein quadric.

###  3.4. Bruhat decomposition and induced coordinates

So far, we described the coadjoint orbits in terms of matrices and gave a
geometrical interpretation in terms of flag manifolds. Now we will focus on
the analytical structure which will allow us to determine the Laplace operator
to the aim of finding the corresponding Green’s function. For that, we need a
bit of notation:

A closed subgroup $P$ of a Lie group $G$ is parabolic if the quotient variety
$G/P$ satisfies the following property: for any variety $Y$, the projection
map $(G/P)\times Y\to Y$ maps closed set to closed sets. Furthermore, a
closed, connected and solvable subgroup of $G$ is called a Borel subgroup.
Note that all Borel subgroups are mutually conjugate.

Given a Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, its complexification is
defined by
$\mathfrak{g}^{\mathbb{C}}:=\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}.$
Moreover, recall that a real subalgebra $\mathfrak{f}$ of a complex Lie
algebra $\mathfrak{h}$ is called a real form of $\mathfrak{h}$ if every
$h\in\mathfrak{h}$ can be uniquely written as $h=h_{1}+ih_{2}$ with
$h_{1},h_{2}\in\mathfrak{f}$. The complexification of $\mathfrak{f}$ yields
again $\mathfrak{h}$, i.e., $\mathfrak{f}^{\mathbb{C}}\cong\mathfrak{h}$. Note
that not every complex Lie algebra has a real form. Moreover, there are in
general several non-isomorphic real forms for a given complex Lie algebra:

######  Example 3.6.

The Lie algebra $\mathfrak{sl}(3,\mathbb{C})$ has the following (non-
isomorphic) real forms:

$\displaystyle\mathfrak{sl}(3,\mathbb{R})$
$\displaystyle=\left\\{\left.\begin{pmatrix}a&b&c\\\ d&e&f\\\
g&h&-(a+e)\end{pmatrix}\right|a,b,c,d,e,f,g,h\in\mathbb{R}\right\\},$
$\displaystyle\mathfrak{su}(1,2)$
$\displaystyle=\left\\{\left.\begin{pmatrix}a+bi&c+di&ei\\\ f+gi&-2bi&-c+di\\\
hi&-f+gi&-a+bi\end{pmatrix}\right|a,b,c,d,e,f,g,h\in\mathbb{R}\right\\},$
$\displaystyle\mathfrak{su}(3)$
$\displaystyle=\left\\{\left.\begin{pmatrix}ai&c+di&g+hi\\\ -(c-di)&ib&e+fi\\\
-(g-hi)&-(e-fi)&-i(a+b)\end{pmatrix}\right|a,b,c,d,e,f,g,h\in\mathbb{R}\right\\}.$

The first one is called the split real form, the second one quasi-split form,
and the last one is referred to as compact form.

Let $G$ be a compact and connected Lie group. The complexification of $G$ is
defined as the complex Lie group $G^{\mathbb{C}}$ that contains $G$ as a
closed subgroup and that has the following (universal) property: every
homomorphism $f:G\to L$, for every complex Lie group $L$, lifts to a
homomorphism $G^{\mathbb{C}}\to L$. Moreover, on the level of Lie algebras,
$\mathsf{Lie}(G^{\mathbb{C}})=\mathfrak{g}^{\mathbb{C}}$ is the
complexification of $\mathsf{Lie}(G)=\mathfrak{g}$.

The following result was proven in more generality by Picken [Pic90], but we
sketch the proof here for the reader’s convenience.

######  Lemma 3.7.

Denote by $B$ the subgroup of upper triangular matrices of
$\mathsf{SL}(3,\mathbb{C})$. Then there is an isomorphism
$\mathsf{SU}(3)/\mathbb{T}^{2}\cong\mathsf{SL}(3,\mathbb{C})/B$.

###### Proof.

Take a matrix $g\in\mathsf{SL}(3,\mathbb{C})$ and denote its columns by
$\underline{g}_{k}$ for $k=1,2,3$ so that the matrix can be written as
$g=(\underline{g}_{1},\underline{g}_{2},\underline{g}_{3})$. Let
$\langle\cdot,\cdot\rangle_{H}$ be the Hermitian inner product on
$\mathbb{C}^{3}$ given by $\langle
x,y\rangle_{H}=x_{1}\bar{y}_{1}+x_{2}\bar{y}_{2}+x_{3}\bar{y}_{3}$ for
$x=(x_{1},x_{2},x_{3}),y=(y_{1},y_{2},y_{3})\in\mathbb{C}^{3}$. A priori, the
vectors $\underline{g}_{k}$ are not orthonormal. Nevertheless, using the
Gramm-Schmidt procedure they can be made orthonormal:

$\displaystyle\underline{g}_{1}^{\prime}:$ $\displaystyle=\underline{g}_{1}$
$\displaystyle\underline{g}_{2}^{\prime}:$
$\displaystyle=\underline{g}_{2}-\frac{\langle\underline{g}_{2},\underline{g}_{1}^{\prime}\rangle}{\langle\underline{g}_{1}^{\prime},\underline{g}_{1}^{\prime}\rangle}\underline{g}_{1}^{\prime}$
$\displaystyle\underline{g}_{3}^{\prime}:$
$\displaystyle=\underline{g}_{3}-\frac{\langle\underline{g}_{3},\underline{g}_{2}^{\prime}\rangle}{\langle\underline{g}_{2}^{\prime},\underline{g}_{2}^{\prime}\rangle}\underline{g}_{2}^{\prime}-\frac{\langle\underline{g}_{3},\underline{g}_{1}^{\prime}\rangle}{\langle\underline{g}_{1}^{\prime},\underline{g}_{1}^{\prime}\rangle}\underline{g}_{1}^{\prime}$

Normalising each of them via
$\underline{v}_{k}:=\frac{\underline{g}_{k}^{\prime}}{||\underline{g}_{k}^{\prime}||}$,
the matrix given by
$U=(\underline{v}_{1},\underline{v}_{2},\underline{v}_{3})$ is an element of
$\mathsf{SU}(3)$. Moreover, it satisfies $U=gb^{\prime}$ for some upper
triangular matrix $b^{\prime}\in B$ which performs the Gramm-Schmidt procedure
on $g$. Thus $g=U(b^{\prime})^{-1}$ lies in $\mathsf{SL}(3,\mathbb{C})/B$.
This means that we can write $\mathsf{SL}(3,\mathbb{C})=\mathsf{SU}(3)B$ and
that $\mathsf{SU}(3)\cap B=\mathbb{T}^{2}$ (this intersection exactly result
in the 2-torus). This induces the wanted isomorphism in the following way:
take an equivalence class $[g]_{B}\in\mathsf{SL}(3,\mathbb{C})/B$ which can
also be written as $[Ub]_{B}$ (see Gramm-Schmidt procedure). It is then mapped
to $[U]_{\mathbb{T}^{2}}\in\mathsf{SU}(3)/\mathbb{T}^{2}$. ∎

The power of this isomorphism lies in the fact that one can make the
transition from the geometrical picture of the coadjoint orbit (as being the
flag manifold realised as the homogeneous space
$\mathsf{SU}(3)/\mathbb{T}^{2}$) to the complex manifold
$\mathsf{SL}(3,\mathbb{C})/B$. This is convenient as there exists a well-
developed theory of so-called Bruhat coordinates on the complex manifold,
which will be useful for our approach.

######  Definition 3.8.

Let $\mathfrak{g}$ be a semi-simple Lie algebra with Cartan algebra
$\mathfrak{h}$ and root system $\Phi$. The weight space decomposition of a Lie
algebra $\mathfrak{g}$ is given by the direct sum decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$
where $\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}\mid[H,X]=\alpha(H)\text{ for
}H\in\mathfrak{h}\\}$. All $\alpha\in\mathfrak{h}^{*}$ that are non zero are
called roots.

Recall that a Lie algebra is simple if $\mathfrak{g}$ is not abelian and
$\mathfrak{g}$ has no non-trivial ideals. For a simple Lie algebra
$\mathfrak{g}$, we have the triangular decomposition

$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$

where $\mathfrak{h}$ is the Cartan subalgebra and
$\mathfrak{n}_{\pm}:=\bigoplus_{\alpha\in\Phi^{\pm}}\mathfrak{g}_{\alpha}$ are
the so-called upper and lower nilpotent subalgebras consisting of the positive
(resp. negative) roots of $\mathfrak{g}$. Moreover, set

$\mathfrak{b}_{\pm}:=\mathfrak{h}\oplus\mathfrak{n}_{\pm}$

and call them the upper and lower Borel subalgebras. On the Lie group level,
$B^{\pm}$ and $N^{\pm}$ are called the Borel subgroups and unipotent subgroups
of the Lie group $G$. In particular, we say that $N^{-}$ is the opposite
unipotent subgroup.

Let $G$ be a semisimple Lie group with Lie algebra $\mathfrak{g}$. Consider a
Borel subgroup $B\leq G$ and the Weyl group $\mathcal{W}$ associated with $G$.
Then the Bruhat decomposition of $G$ is given by
$G=\bigcup_{w\in\mathcal{W}}BwB$. This decomposition gives rise to the cell
decomposition of the homogeneous space $G/B=\bigcup_{w\in\mathcal{W}}BwB/B$.
Each of the $BwB/B$ corresponds to an affine space of dimension $\ell(w)$
where $\ell(w)$ is the length of the Weyl group element $w$ given by the
minimal $k$ such that $w$ can be written as a product of $k$ generators of the
Weyl group. Note that there is always an element in the Weyl group
$\mathcal{W}$ which has maximal length, in this case the length of this
element equals the number of positive roots $\Phi^{+}$. This element in the
Weyl group is denoted by $w_{o}\in\mathcal{W}$. It has the property that
$w_{0}(\Phi^{+})=\Phi^{-}$, i.e. it interchanges the positive and negative
roots. In the Bruhat decomposition

$G/B=\bigcup_{w\in\mathcal{W}}BwB/B,$

the identity element $e$ gives rise to an open subset $BeB$ which is called
the big cell and is denoted by $X_{e}$. The flag manifold is identified with
the space $\mathsf{SL}(3,\mathbb{C})/B$ where $B$ is the subgroup of upper
triangular matrices. In this context, we have

$\displaystyle N^{-}=\left\\{\left.\begin{pmatrix}1&0&0\\\ z_{1}&1&0\\\
z_{2}&z_{3}&1\end{pmatrix}\right|z_{1},z_{2},z_{3}\in\mathbb{C}\right\\}.$

######  Proposition 3.9 ([BMHM94]).

$N^{-}$ acts freely and transitively on the big cell $X_{e}$. Thus we may
identify $X_{e}$ with $N^{-}$.

Note that the translates $g\cdot N_{-}$ of the big cell (under the $G$-action)
cover the whole flag manifold.

The flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ can also be seen as

$\displaystyle\mathbb{F}_{1,2}(\mathbb{C}^{3})=\\{(V_{1},V_{2})\in\mathbb{CP}^{2}\times(\mathbb{CP}^{2})^{*}\mid
V_{2}(V_{1})=0\\}.$ (3.4.1)

This means intuitively that the manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$
consists of all pairs $(V_{1},V_{2})$ where $V_{2}$ is a projective line in
$\mathbb{CP}^{2}$ and $V_{1}$ is a point on the line.

######  Theorem 3.10 ([BMHM94]).

Consider the group $\mathsf{SL}(3,\mathbb{C})$ and the Borel subgroup $B$ of
upper triangular matrices. Let $R_{1}:=[0:0:1]\in\mathbb{CP}^{2}$ and
$R_{2}:=[0:x:y]\in(\mathbb{CP}^{2})^{*}$. Then the (Bruhat) cell-decomposition
of $\mathsf{SL}(3,\mathbb{C})/B$ consists of the following six cells (where
the indices in the Bruhat cell correspond to the group elements of the
symmetric group in three elements, see Table 1):

  1. (1)

The big cell, which has codimension zero, is given by

$\displaystyle X_{e}:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
R_{2}(V_{1})\neq 0,\ V_{2}(R_{1})\neq 0\\}.$

  2. (2)

There are two Bruhat cells of codimension one given by

$\displaystyle X_{(1,2)}$
$\displaystyle:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
R_{2}(V_{1})=0,\ R_{1}\neq V_{1},\ R_{2}\neq V_{2}\\},$ $\displaystyle
X_{(2,3)}$
$\displaystyle:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
V_{2}(R_{1})=0,\ R_{1}\neq V_{1},\ R_{2}\neq V_{2}\\}.$

  3. (3)

There are two Bruhat cells of codimension two given by

$\displaystyle X_{(1,2,3)}$
$\displaystyle:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
V_{1}=R_{1},\ R_{2}\neq V_{2}\\},$ $\displaystyle X_{(1,3,2)}$
$\displaystyle:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
V_{2}=R_{2},\ R_{1}\neq V_{1}\\}.$

  4. (4)

The $0$-cell, which has codimension three, is given by

$\displaystyle
X_{(1,3)}:=\\{(V_{1},V_{2})\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\mid
V_{1}=R_{1},\ R_{2}=V_{2}\\}.$

Moreover, the opposite unipotent subgroup $N^{-}$ acts transitively on the big
cell $X$.

Recall that the Weyl group of $\mathsf{SL}(3,\mathbb{C})$ is isomorphic to
$\mathsf{Sym}(3)$, the symmetric group of order three. We now give an overview
how the elements of $\mathsf{Sym}(3)$ correspond to the elements of the Weyl
group and their associated Bruhat decomposition and Weyl length.

Group element | Bruhat expression | Length | Matrix representation
---|---|---|---
$e$ | empty word | 0 | $\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$
$(1,2)$ | $s_{1}$ | 1 | $\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}$
$(2,3)$ | $s_{2}$ | 1 | $\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&-1&0\end{pmatrix}$
$(1,2,3)$ | $s_{1}s_{2}$ | 2 | $\begin{pmatrix}0&0&-1\\\ 1&0&0\\\ 0&-1&0\end{pmatrix}$
$(1,3,2)$ | $s_{2}s_{1}$ | 2 | $\begin{pmatrix}0&-1&0\\\ 0&0&1\\\ -1&0&0\end{pmatrix}$
$(1,3)$ | $s_{1}s_{2}s_{1}$ | 3 | $\begin{pmatrix}0&0&-1\\\ 0&-1&0\\\ -1&0&0\end{pmatrix}$
Table 1. Bruhat expressions.

To work on the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$, we need
explicit coordinates.

######  Corollary 3.11.

The big cell of the flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{3})=\mathsf{SL}(3,\mathbb{C})/B$ can be
identified with $N^{-}$. This leads to the following coordinate chart for the
big cell of $\mathbb{F}_{1,2}(\mathbb{C}^{3})$:

$\displaystyle
N^{-}\to\mathbb{C}^{3}\simeq\mathbb{R}^{6},\quad\begin{pmatrix}1&0&0\\\
z_{1}&1&0\\\
z_{2}&z_{3}&1\end{pmatrix}\mapsto(z_{1},z_{2},z_{3})\simeq(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}).$
(3.4.2)

###  3.5. The Kähler structure of coadjoint orbits

Let $M$ be a complex manifold of complex dimension $n$ with local coordinates
$(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$. Then a Hermitian metric $h$ is of the
form

$\displaystyle h=\sum_{i,j=1}^{n}h_{ij}dz_{i}\otimes d\overline{z}_{j}$

where $(h_{ij})_{1\leq i,j\leq n}$ is a positive-definite Hermitian matrix. A
complex manifold $M$ equipped with a Hermitian metric $h$ is called a
Hermitian manifold. A Hermitian manifold $(M,h)$ carries a natural symplectic
form, more precisely, the $(1,1)$-form given by imaginary part of the
Hermitian metric $h$ is symplectic and has the explicit expression

$\displaystyle\omega$
$\displaystyle:=-\Im(h)=-\frac{1}{2i}(h-\overline{h})=\frac{i}{2}\sum_{1\leq
i,j\leq n}h_{ij}dz_{i}\otimes
d\overline{z}_{j}-h_{ji}d\overline{z}_{i}\otimes{dz}_{j}$
$\displaystyle=\frac{i}{2}\sum_{1\leq i,j\leq n}h_{ij}(dz_{i}\otimes
d\overline{z}_{j}-d\overline{z}_{j}\otimes dz_{i})=\frac{i}{2}\sum_{1\leq
i,j\leq n}h_{ij}dz_{i}\wedge d\overline{z}_{j}.$

This $\omega$ is often referred to as the fundamental form on $(M,h)$.

An almost complex structure $J$ on a smooth manifold $M$ is an isomorphism
$J:TM\to TM$ with $J^{2}=-\operatorname{Id}$. Such a $J$ is integrable if the
so-called Nijenhuis tensor

$N_{J}(X,Y):=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]$

vanishes for all vector fields $X,Y$ on the manifold $M$.

A symplectic manifold $(M,\omega)$ is Kähler if there exists an integrable
almost complex structure $J$ such that the bilinear form
$g(u,v):=\omega(u,Jv)$ is symmetric and positive definite for all $u,v\in TM$,
i.e., $g$ is a Riemannian metric.

A Hermitian manifold $(M,h)$ resp. $h$ is Kähler if the fundamental form
$-\Im(h)$ is closed, i.e. $-d\Im(h)=0$. Moreover, in this situation, $-\Im(h)$
is in fact a (real) symplectic form on $(M,h)$.

######  Lemma 3.12.

Let $(M,h)$ be a Kähler manifold. Then for all $p\in M$ there exists a open
neighbourhood $U$ of $p$ and a function $K_{U}:U\to\mathbb{R}$ such that

$h_{ij}=\partial_{z_{i}}\partial_{\bar{z_{j}}}K_{U}(z_{i},\bar{z_{j}})\quad\mbox{for
all }1\leq i,j\leq n$

for local complex coordinates $z$ on $U$. This locally defined function is
usually called the Kähler potential and denoted by $K_{M}$.

Note that $(h_{ij})_{1\leq i,j\leq n}$ is defined globally, whereas the
potential is only defined locally. Using the Dolbeault operators

$\partial:=\sum_{k=1}^{n}\partial_{z_{k}}dz_{k}\qquad\text{and}\qquad\overline{\partial}:=\sum_{k=1}^{n}\partial_{\overline{z}_{k}}d\overline{z}_{k},$

the fundamental $(1,1)$-form can be expressed as
$\omega=i\partial\overline{\partial}K_{M}$.

######  Example 3.13.

On $\mathbb{R}^{2n}\cong\mathbb{C}^{n}$, consider the Euclidean metric
$g_{E}$, the standard symplectic form $\omega_{st}$, and standard compatible
complex structure $J_{st}$ given in matrix notation by

$\displaystyle g_{E}=\begin{pmatrix}I_{n}&0\\\ 0&I_{n}\end{pmatrix},$
$\displaystyle\omega_{st}=\begin{pmatrix}0&I_{n}\\\ -I_{n}&0\end{pmatrix},$
$\displaystyle J_{st}=\begin{pmatrix}0&-I_{n}\\\ I_{n}&0\end{pmatrix}$

where $I_{n}$ is the $(n\times n)$-unit matrix. Then
$K_{\mathbb{C}^{n}}:\mathbb{C}^{n}\to\mathbb{R}$ given by
$K_{\mathbb{C}^{n}}(z):=\frac{|z|}{2}$ is a Kähler potential since

$\displaystyle
i\partial\overline{\partial}\left(\frac{|z|}{2}\right)=\frac{i}{2}\partial\overline{\partial}\sum_{k=1}^{n}z_{k}\overline{z}_{k}=\frac{i}{2}\sum_{k=1}^{n}dz_{k}\wedge
d\overline{z}_{k}.$

An important class of Kähler manifolds is given by coadjoint orbits:

######  Theorem 3.14 (Bott).

Let $G$ be a semi-simple compact Lie group. Each (co)adjoint orbit has a
$G$-equivariant Kähler structure.

Later on, we will study the point vortex dynamics modelled on the degenerate
orbit $\mathbb{CP}^{2}$ and the generic orbit given by flag manifold
${F}_{1,2}(\mathbb{C}^{3})$. Therefore it is useful to know their Kähler
potentials.

######  Lemma 3.15 (Picken [Pic90]).

The Kähler potentials on $\mathbb{CP}^{n}$ and on the flag manifold are given
by the following logarithmic functions depending on local coordinates
$(1:z_{1}:z_{2}:\cdots:z_{n})$ on $\mathbb{CP}^{n}$ and the local coordinates
from Corollary 3.11 for the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$.

$\displaystyle K_{\mathbb{CP}^{n}}$
$\displaystyle=\log\left(1+\sum_{k=1}^{n-1}|z_{k}|^{2}\right),$ (3.5.1)
$\displaystyle K_{{\mathbb{F}_{1,2}(\mathbb{C}^{3})}}$
$\displaystyle=\log\left((1+|z_{1}|^{2}+|z_{2}|^{2})(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2})\right)=:\log(K_{1}K_{2}).$
(3.5.2)

Moreover,

######  Lemma 3.16 (Muñoz $\&$ González-Prieto $\&$ Rojo [MGPR20]).

In homogeneous coordinates $(1:z_{1}:z_{2}:\cdots:z_{n})\in\mathbb{CP}^{n}$,
the Hermitian metric $h=(h_{ij})_{1\leq i,j\leq n}$ on $\mathbb{CP}^{n}$ takes
the following form:

$h_{ij}=\frac{(1+|z_{1}|^{2}+|z_{2}|^{2})\delta_{ij}-\overline{z}_{i}z_{j}}{(1+|z_{1}|^{2}+|z_{2}|^{2})^{2}}=\frac{(1+|z|^{2})\delta_{ij}-\overline{z}_{i}z_{j}}{(1+|z|^{2})^{2}}$

where $\delta_{ij}$ is the Kronecker symbol. Written as matrix, we have

$(h_{ij})_{ij}={\frac{1}{(1+|{z}|^{2})^{2}}}{\begin{pmatrix}1+|{z}|^{2}-|z_{1}|^{2}&-{\bar{z}}_{1}z_{2}&\cdots&-{\bar{z}}_{1}z_{n}\\\
-{\bar{z}}_{2}z_{1}&1+|{z}|^{2}-|z_{2}|^{2}&\cdots&-{\bar{z}}_{2}z_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
-{\bar{z}}_{n}z_{1}&-{\bar{z}}_{n}z_{2}&\cdots&1+|{z}|^{2}-|z_{n}|^{2}\end{pmatrix}}.$

Its determinant is given by $\det(h_{ij})=\frac{1}{(1+|z|^{2})^{n+1}}$.

######  Lemma 3.17.

Recall from Lemma 3.15 the real valued functions

$\displaystyle K_{1}=1+|z_{1}|^{2}+|z_{2}|^{2}\quad\mbox{and}\quad
K_{2}=1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}.$

Then the Hermitian metric $(h_{ij})_{1\leq i\leq j\leq 3}$ on
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$ has the following matrix representation:

$(h_{ij})_{1\leq i,j\leq
n}=\begin{pmatrix}\frac{1+|z_{2}|^{2}}{K_{1}^{2}}+\frac{|z_{3}|^{2}(1+|z_{3}|^{2})}{K_{2}^{2}}&-\frac{\overline{z}_{1}z_{2}}{K_{1}^{2}}-\frac{z_{3}(1+|z_{3}|^{2})}{K_{2}^{2}}&\frac{z_{3}(\overline{z}_{1}+\overline{z}_{2}z_{3})}{K_{2}^{2}}\\\
-\frac{{z}_{1}\overline{z}_{2}}{K_{1}^{2}}-\frac{\overline{z}_{3}(1+|{z}_{3}|^{2})}{K_{2}^{2}}&\frac{1+|z_{1}|^{2}}{K_{1}^{2}}+\frac{1+|z_{3}|^{2}}{K_{2}^{2}}&-\frac{\overline{z}_{1}+\overline{z}_{2}z_{3}}{K_{2}^{2}}\\\
\frac{{\overline{z}}_{3}({z}_{1}+{z}_{2}\overline{z}_{3})}{K_{2}^{2}}&-\frac{{z}_{1}+{z}_{2}\overline{z}_{3}}{K_{2}^{2}}&\frac{K_{1}}{K_{2}^{2}}\end{pmatrix}.$

Its determinant is given by $\det\bigl{(}(h_{ij})_{1\leq i,j\leq
n}\bigr{)}=\frac{2}{K_{1}^{2}K_{2}^{2}}$.

###### Proof.

Recall from Lemma 3.15 the expression for the Kähler potential

$\displaystyle\log(1+|z_{1}|^{2}+|z_{2}|^{2})(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2})=\log
K_{1}K_{2}=\log K_{1}+\log K_{2}.$

The entries of the matrix $(h_{ij})_{1\leq i\leq j\leq 3}$ are computed via
$h_{ij}=\partial_{z_{i}}\partial_{\bar{z_{j}}}K_{M}(z_{i},\bar{z_{j}})$.
Exemplarily we now compute the entry

$h_{11}=\partial_{z_{1}}\partial_{\overline{z}_{1}}\log((1+|z_{1}|^{2}+|z_{2}|^{2}))+\partial_{z_{1}}\partial_{\overline{z}_{1}}\log(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}).$

The first term in this expression becomes

$\displaystyle\partial_{z_{1}}\frac{z_{1}}{1+|z_{1}|^{2}+|z_{2}|^{2}}=\frac{(1+|z_{1}|^{2}+|z_{2}|^{2})-z_{1}\overline{z}_{1}}{(1+|z_{1}|^{2}+|z_{2}|^{2})^{2}}=\frac{1+|z_{2}|^{2}}{K_{1}^{2}}$

and the second one

$\displaystyle\partial_{z_{1}}\frac{\overline{z}_{3}(z_{1}z_{3}-z_{2})}{1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}}$
$\displaystyle=\frac{|z_{3}|^{2}(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2})-|z_{3}|^{2}|z_{1}z_{3}-z_{2}|^{2}}{(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2})^{2}}$
$\displaystyle=\frac{|z_{3}|^{2}(1+|z_{3}|^{2})}{K_{2}^{2}}.$

Altogether, we obtain

$h_{11}=\frac{1+|z_{2}|^{2}}{K_{1}^{2}}+\frac{|z_{3}|^{2}(1+|z_{3}|^{2})}{K_{2}^{2}}.$

The other entries are computed analogously. ∎

A straightforward computation yields:

######  Corollary 3.18.

The inverse matrix $((h_{ij})_{1\leq i\leq j\leq 3})^{-1}=:(h^{ij})_{1\leq
i\leq j\leq 3}$ is given by:

$\displaystyle{\footnotesize\begin{pmatrix}K_{1}\left(1+|z_{1}|^{2}+\frac{K_{1}}{K_{2}}\right)&K_{1}\left(\overline{z}_{1}z_{2}+\frac{K_{1}}{K_{2}}{z}_{3}\right)&(\overline{z}_{1}+z_{3}\overline{z}_{2})(\overline{z}_{1}z_{2}-z_{3}-z_{3}|z_{1}|^{2})\\\
K_{1}\left(z_{1}\overline{z}_{2}+\frac{K_{1}}{K_{2}}\overline{z}_{3}\right)&K_{1}\left((1+|z_{2}|^{2})+\frac{K_{1}}{K_{2}}|z_{3}|^{2}\right)&(\overline{z}_{1}+\overline{z}_{2}z_{3})\left((1+|z_{2}|^{2})-z_{1}\overline{z}_{2}z_{3}\right)\\\
(z_{1}+\overline{z}_{3}z_{2})(z_{1}\overline{z}_{2}-\overline{z}_{3}-z_{3}|z_{1}|^{2})&({z}_{1}+{z}_{2}\overline{z}_{3})\left((1+|z_{2}|^{2})-{\overline{z}}_{1}{z}_{2}\overline{z}_{3}\right)&K_{1}(1+|z_{3}|^{2})+\frac{K_{2}^{2}}{K_{1}}\end{pmatrix}}$

where $K_{1}=1+|z_{1}|^{2}+|z_{2}|^{2}$ and
$K_{2}=1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}$.

###  3.6. Different symplectic structures

Important for us is the following result due to Kirillov, Kostant and Souriau:

######  Theorem 3.19.

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra with dual
$\mathfrak{g}^{*}$ and $\mu\in\mathfrak{g}^{*}$. Then the coadjoint orbit
$\mathcal{O}_{\mu}$ carries the canonical symplectic form

$\displaystyle\omega^{KKS}_{\nu}(\operatorname{ad}_{\xi}^{*}\nu,\operatorname{ad}_{\eta}^{*}\nu):=\langle\nu,[\xi,\eta]\rangle,$

where $\xi,\eta\in\mathfrak{g}$ and $\nu\in\mathcal{O}_{\mu}$. This symplectic
form is usually called the Kirillov-Kostant-Souriau (KKS) symplectic form.

This implies that, considered as coadjoint orbit, the flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{3})=\mathsf{SU}(3)/\mathbb{T}^{2}$ can be
endowed with $\omega^{KKS}$ as symplectic form. Note that there is an
additional way to consider $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ as symplectic
manifold: We consider the complexification of $\mathsf{SU}(3)/\mathbb{T}^{2}$
given by
$\left(\mathsf{SU}(3)/\mathbb{T}^{2}\right)^{\mathbb{C}}=\mathsf{SL}(3,\mathbb{C})/B$.
Using the coordinates on the big cell given in (3.4.2) and the Hermitian
metric from Lemma 3.17, Picken $\&$ Duistermaat [Pic90] give the following
formula for the symplectic form on $(W^{6})^{\mathbb{C}}$:

$\displaystyle\omega^{(W^{6})^{\mathbb{C}}}:=\frac{i}{2}\left(\partial\overline{\partial}\log\left(1+\sum_{k=1}^{2}|z_{k}|^{2}\right)+\partial\overline{\partial}\log\left(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}\right)\right).$

Note that, if $\omega^{\mathbb{CP}^{2}}$ denotes the Fubini-Study form on
$\mathbb{CP}^{2}$ then the symplectic form $\omega^{(W^{6})^{\mathbb{C}}}$
consists of the Fubini-Study form
$\omega^{\mathbb{CP}^{2}}=\frac{i}{2}\partial\overline{\partial}\log\left(1+\sum_{k=1}^{2}|z_{k}|^{2}\right)$
on $\mathbb{CP}^{2}$ plus the correction term
$\tilde{\omega}=\frac{i}{2}\partial\overline{\partial}\log\left(1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}\right),$
i.e., $\omega^{(W^{6})^{\mathbb{C}}}=\omega^{\mathbb{CP}^{2}}+\tilde{\omega}$.

######  Remark 3.20 (Picken $\&$ Duistermaat [Pic90], Bernatska $\&$ Holod et
al. [BH08]).

$\left((W^{6})^{\mathbb{C}},\ \omega^{(W^{6})^{\mathbb{C}}}\right)$ and
$\left(\mathbb{F}_{1,2}(\mathbb{C}^{3})=\mathsf{SU}(3)/\mathbb{T}^{2},\
\omega^{KKS}\right)$ are symplectomorphic.

Summarized, we have the following types of coadjoint orbits of
$\mathsf{SU}(3)$, each endowed with its natural symplectic structure:

coadjoint orbit | symplectic form | (real) dimension
---|---|---
$\frac{\operatorname{\mathsf{SU}}(3)}{\operatorname{\mathsf{U}}(1)\times\operatorname{\mathsf{U}}(1)}$ | Kirillov-Kostant-Souriau | 6
$\mathbb{CP}^{2}$ | Fubini-Study | 4
point | trivial | 0
Table 2. Coadjoint orbits of $\operatorname{\mathsf{SU}}(3)$.

##  4\. The point vortex momentum map on $\mathbb{CP}^{2}$ and
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$

In this section, we will study the Hamiltonian action of $\mathsf{SU}(3)$ on
(products of) coadjoint orbits. In the case of the degenerate orbit, the
dynamics have been studied before: for example, the Hamiltonian action of
$\mathsf{SU}(3)$ on $\mathbb{CP}^{2}\times\mathbb{CP}^{2}$ has been studied by
Beddulli $\&$ Gori [BG07]. Moreover, Montaldi $\&$ Shaddad [MS19a] considered
a similar problem but added a copy of the projective plane. To be more
precise, they considered the (diagonal) action of $\mathsf{SU}(3)$ on
$\mathbb{CP}^{2}\times\mathbb{CP}^{2}\times\mathbb{CP}^{2}$ and the associated
properties of the (weighted) momentum map. We will focus on the generic orbit,
which is the six-dimensional flag manifold and construct a momentum map
$\mu:\mathcal{O}^{\mathsf{SU}(3)}\to\mathfrak{su}(3)^{*}$ explicitly.

###  4.1. The momentum map for vortex dynamics

Let $N\in\mathbb{N}$ and, for $1\leq k\leq N$, let
$\Gamma_{k}\in\mathbb{R}^{\neq 0}$ (‘weight’) and let $(M_{k},\omega_{k})$ be
a symplectic manifold. Let $G$ be a Lie group that acts on each
$(M_{k},\omega_{k})$ with momentum map $\mu_{k}:M_{k}\to\mathfrak{g}^{*}$. Now
set $M:=\Pi_{k=1}^{N}M_{k}$ and equip it with the weighted symplectic form
$\omega_{M}:=\sum_{k=1}^{N}\Gamma_{k}\tau_{k}^{*}\omega_{k}$ where
$\tau_{k}:M\to M_{k}$ is the projection on the $k$th factor. The diagonal
action of $G$ on $M$ is given by $g.m:=(g.m_{1},\dots,g.m_{N})$ for $g\in G$
and $m=(m_{1},\dots,m_{N})\in M$ and its momentum map is given by

$\displaystyle\mu_{M}:M\to\mathfrak{g}^{*},\qquad\mu_{M}(m_{1},\dots,m_{N})=\sum_{k=1}^{N}\Gamma_{k}\mu_{k}(m_{k}).$

We are interested in the special situation when the symplectic manifolds
$M_{k}$ are coadjoint orbits i.e.
$(M_{k},\omega_{k})=(\mathcal{O},\omega_{\mathcal{O}})$. In the next
subsections, we study momentum maps of vortex dynamics for the following two
situations:

$\displaystyle\mu_{\mathcal{O}_{d}^{\mathsf{SU}(3)}}:\mathcal{O}_{d}^{\mathsf{SU}(3)}\simeq\mathbb{CP}^{2}\to\mathfrak{su}(3)^{*}\quad\text{and}\quad\mu_{\mathcal{O}^{\mathsf{SU}(3)}}:\mathcal{O}^{\mathsf{SU}(3)}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})\to\mathfrak{su}(3)^{*}.$

Recall that we identify $\mathfrak{su}(3)$ with $\mathfrak{su}(3)^{*}$ using
the Killing form. Thus, $\mathfrak{su}(3)\cong\mathfrak{su}(3)^{*}$ is
identified with the space of complex skew-Hermitian matrices with trace zero.

###  4.2. The momentum map of the degenerate orbit
$\mathcal{O}_{d}^{\mathsf{SU}(3)}\simeq\mathbb{CP}^{2}$

In this subsection, we recall some facts from Montaldi $\&$ Shaddad [MS19b]
concerning the momentum map of the degenerate coadjoint orbit
$\mathcal{O}_{d}^{\mathsf{SU}(3)}\simeq\mathbb{CP}^{2}$ of $\mathsf{SU}(3)$.

######  Theorem 4.1 (Montaldi & Shaddad [MS19b]).

The momentum map for the Fubini-Study form on $\mathbb{CP}^{2}$ is given by

$\displaystyle\mu:\mathbb{CP}^{2}\to\mathfrak{su}(3)^{*},\quad[x:y:z]\mapsto\begin{pmatrix}|x|^{2}-\frac{1}{3}&x\overline{y}&x\overline{z}\\\
\overline{x}y&|y|^{2}-\frac{1}{3}&y\overline{z}\\\
\overline{x}z&\overline{y}z&|z|^{2}-\frac{1}{3}\end{pmatrix}.$

Furthermore, the map satisfies the following properties:

  1. (i)

$\mu$ is $\mathsf{SU}(3)$-equivariant for the left action, i.e.
$\mu(gZ)=g\mu(Z)$ for all $g\in\operatorname{\mathsf{SU}}(3)$ and all
$Z\in\mathbb{CP}^{2}$.

  2. (ii)

The image of $\mu$ consists of $(3\times 3)$ Hermitian matrices with
eigenvalues $-\frac{1}{3},-\frac{1}{3}$ and $\frac{2}{3}$.

###### Proof.

We briefly sketch a part of the proof: the characteristic polynomial $\chi(u)$
of the matrix $\mu(x,y,z)$ is given by

$\chi(u)=(|x|^{2}+|y|^{2}+|z|^{2})\left(u^{2}+\frac{2}{3}u+\frac{1}{9}\right)-u^{3}-u^{2}-\frac{u}{3}-\frac{1}{27}.$

Solving the equation $\chi(u)=0$ and using the fact that
$|x|^{2}+|y|^{2}+|z|^{2}=1$ gives the three eigenvalues $u_{1}=-\frac{1}{3}$,
$u_{2}=-\frac{1}{3}$ and $u_{3}=\frac{2}{3}$. ∎

###  4.3. The momentum map of the generic orbit
$\mathcal{O}^{\mathsf{SU}(3)}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})$

In order to obtain the momentum map on the flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{3})\simeq\mathcal{O}^{\mathsf{SU}(3)}$ we need
to have the infinitesimal generators of the Lie algebra $\mathfrak{su}(3)$ at
our disposal. They are provided by the following statement:

######  Lemma 4.2.

Let $\lambda_{1},\dots,\lambda_{8}$ be the rescaled basis from Notation 2.1.
Then the infinitesimal vector fields of the Lie algebra $\mathfrak{su}(3)$ on
the flag manifold are given by

$\displaystyle X_{\lambda_{1}}$
$\displaystyle=\frac{i}{2}\left((1-z_{1}^{2})\partial_{z_{1}}-z_{1}z_{2}\partial_{z_{2}}+(z_{1}z_{3}-z_{2})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{2}}$
$\displaystyle=\frac{1}{2}\left((-z_{1}^{2}-1)\partial_{z_{1}}-z_{1}z_{2}\partial_{z_{1}}+(z_{1}z_{3}-z_{2})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{3}}$
$\displaystyle=\frac{i}{2}\left(-2z_{1}\partial_{z_{1}}-z_{2}\partial_{z_{2}}+z_{3}\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{4}}$
$\displaystyle=\frac{i}{2}\left(-z_{1}z_{2}\partial_{z_{1}}+(1-z_{2}^{2})\partial_{z_{2}}-z_{3}(z_{2}-z_{1}z_{3})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{5}}$
$\displaystyle=\frac{1}{2}\left(-z_{1}z_{2}\partial_{z_{1}}-(1+z_{2}^{2})\partial_{z_{2}}-z_{3}(z_{2}-z_{1}z_{3})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{6}}$
$\displaystyle=\frac{i}{2}\left(z_{2}\partial_{z_{1}}+z_{1}\partial_{z_{2}}+(1-z_{3}^{2})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{7}}$
$\displaystyle=\frac{1}{2}\left(z_{2}\partial_{z_{1}}-z_{1}\partial_{z_{2}}-(1-z_{3}^{2})\partial_{z_{3}}\right),$
$\displaystyle X_{\lambda_{8}}$
$\displaystyle=-\frac{i\sqrt{3}}{2}\left(z_{2}\partial_{z_{2}}+z_{3}\partial_{z_{3}}\right).$

###### Proof.

In order to obtain the fundamental vector fields associated to
$\mathfrak{su}(3)$ it is sufficient to determine the vector fields associated
to the basis $\lambda_{1},\dots,\lambda_{8}$ from Notation 2.1. The vector
fields are determined by the equation

$\displaystyle
X_{\lambda_{k}}=\frac{d}{dt}{\bigg{|}_{t=0}}\exp(t\lambda_{k})\cdot\mathcal{Z}$

where
$\mathcal{Z}\in\mathbb{F}_{1,2}(\mathbb{C}^{3})\simeq\mathcal{O}^{\mathsf{SU}(3)}$
(see local coordinates expression from Corollary 3.11) and
$\exp(t\lambda_{k})\cdot\mathcal{Z}$ is defined as multiplication of the
matrices $\exp(t\lambda_{k})$ and $\mathcal{Z}$ and corresponds to the left
action of $\mathsf{SU}(3)$ on the flag manifold.

Without loss of generality, $\mathcal{Z}$ may lie in the big cell and thus is
of the form $\mathcal{Z}=\left(\begin{smallmatrix}1&0&0\\\ z_{1}&1&0\\\
z_{2}&z_{3}&1\end{smallmatrix}\right)$. When $\exp(t\lambda_{k})$ acts on
$\mathcal{Z}$, the result lies not necessarily again in the big cell. But, due
to the fact that the flag manifold is identified with the (complexified)
homogeneous space $\mathsf{SL}(3,\mathbb{C})/B$ where $B$ is the Borel
subgroup of upper triangular matrices, we can always multiply (from the right)
with elements from $B$ to get again an element in the big cell. Using formula
(2.2.1), we compute

$\displaystyle\exp(t\lambda_{1})$
$\displaystyle=\begin{pmatrix}\cos\left(\frac{t}{2}\right)&i\sin\left(\frac{t}{2}\right)&0\\\
i\sin\left(\frac{t}{2}\right)&\cos\left(\frac{t}{2}\right)&0\\\ 0&0&1\\\
\end{pmatrix},$
$\displaystyle\exp(t\lambda_{2})=\begin{pmatrix}\cos\left(\frac{t}{2}\right)&\sin\left(\frac{t}{2}\right)&0\\\
-\sin\left(\frac{t}{2}\right)&\cos\left(\frac{t}{2}\right)&0\\\ 0&0&1\\\
\end{pmatrix},$ $\displaystyle\exp(t\lambda_{3})$
$\displaystyle=\begin{pmatrix}e^{\frac{it}{2}}&0&0\\\ 0&e^{-\frac{it}{2}}&0\\\
0&0&1\\\ \end{pmatrix},$
$\displaystyle\exp(t\lambda_{4})=\begin{pmatrix}\cos\left(\frac{t}{2}\right)&0&i\sin\left(\frac{t}{2}\right)\\\
0&1&0\\\ i\sin\left(\frac{t}{2}\right)&0&\cos\left(\frac{t}{2}\right)\\\
\end{pmatrix},$ $\displaystyle\exp(t\lambda_{5})$
$\displaystyle=\begin{pmatrix}\cos\left(\frac{t}{2}\right)&0&\sin\left(\frac{t}{2}\right)\\\
0&1&0\\\ -\sin\left(\frac{t}{2}\right)&0&\cos\left(\frac{t}{2}\right)\\\
\end{pmatrix},$ $\displaystyle\exp(t\lambda_{6})=\begin{pmatrix}1&0&0\\\
0&\cos\left(\frac{t}{2}\right)&i\sin\left(\frac{t}{2}\right)\\\
0&i\sin\left(\frac{t}{2}\right)&\cos\left(\frac{t}{2}\right)\\\
\end{pmatrix},$ $\displaystyle\exp(t\lambda_{7})$
$\displaystyle=\begin{pmatrix}1&0&0\\\
0&\cos\left(\frac{t}{2}\right)&\sin\left(\frac{t}{2}\right)\\\
0&-\sin\left(\frac{t}{2}\right)&\cos\left(\frac{t}{2}\right)\\\
\end{pmatrix},$
$\displaystyle\exp(t\lambda_{8})=\begin{pmatrix}e^{\frac{it}{2\sqrt{3}}}&0&0\\\
0&e^{\frac{it}{2\sqrt{3}}}&0\\\ 0&0&e^{-\frac{it}{\sqrt{3}}}\\\
\end{pmatrix}.$

Now we need to compute $\exp(t\lambda_{k})\cdot\mathcal{Z}$. Note that, as
mentioned above, the result may not lie in the big cell. Thus we need to
multiply in addition from the right with an element
$\mathbf{b}=\left(\begin{smallmatrix}b_{1}&b_{2}&b_{3}\\\ 0&b_{4}&b_{5}\\\
0&0&b_{6}\end{smallmatrix}\right)\in B$, i.e.,

$\displaystyle\exp(\lambda_{k}t)\begin{pmatrix}1&0&0\\\ z_{1}&1&0\\\
z_{2}&z_{3}&1\end{pmatrix}\begin{pmatrix}b_{1}&b_{2}&b_{3}\\\ 0&b_{4}&b_{5}\\\
0&0&b_{6}\end{pmatrix}$

in order to obtain as element of the big cell

$\displaystyle A_{k}:=\begin{pmatrix}1&0&0\\\ f_{1,k}({z}_{1},t)&1&0\\\
f_{2,k}({z}_{2},t)&f_{3,k}({z}_{3},t)&1\end{pmatrix}$

for some functions $f_{j,k}(z_{j},t)$ with $j\in\\{1,2,3\\}$ and
$k\in\\{1,\dots,8\\}$ depending on the complex variables $z_{j}$ and the real
variable $t$. Now, for $k\in\\{1,\dots,8\\}$, we solve
$\exp(\lambda_{k}t)\mathcal{Z}\mathbf{b}=A_{k}$ for $\mathbf{b}\in B$. The
solutions are denoted by $\beta_{k}\in B$ and are given as follows:

$\displaystyle\beta_{1}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\cos\left(\frac{t}{2}\right)+iz_{1}\sin\left(\frac{t}{2}\right)}&-i\sin\left(\frac{t}{2}\right)&0\\\
0&\cos\left(\frac{t}{2}\right)+iz_{1}\sin\left(\frac{t}{2}\right)&0\\\
0&0&1\\\ \end{array}\right),$ $\displaystyle\beta_{2}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{z_{1}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}&-\sin\left(\frac{t}{2}\right)&0\\\
0&z_{1}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)&0\\\ 0&0&1\\\
\end{array}\right),$ $\displaystyle\beta_{3}$
$\displaystyle=\left(\begin{array}[]{ccc}e^{\frac{it}{2}}&0&0\\\
0&e^{-\frac{it}{2}}&0\\\ 0&0&1\\\ \end{array}\right),$
$\displaystyle\beta_{4}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\cos\left(\frac{t}{2}\right)+iz_{2}\sin\left(\frac{t}{2}\right)}&\frac{z_{3}\sin\left(\frac{t}{2}\right)}{-z_{2}\sin\left(\frac{t}{2}\right)+z_{1}z_{3}\sin\left(\frac{t}{2}\right)+i\cos\left(\frac{t}{2}\right)}&-i\sin\left(\frac{t}{2}\right)\\\
0&1-\frac{z_{1}\left(z_{3}\sin\left(\frac{t}{2}\right)\right)}{-z_{2}\sin\left(\frac{t}{2}\right)+z_{1}z_{3}\sin\left(\frac{t}{2}\right)+i\cos\left(\frac{t}{2}\right)}&iz_{1}\sin\left(\frac{t}{2}\right)\\\
0&0&\cos\left(\frac{t}{2}\right)+i\left(z_{2}-z_{1}z_{3}\right)\sin\left(\frac{t}{2}\right)\\\
\end{array}\right),$ $\displaystyle\beta_{5}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{z_{2}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}&-\frac{z_{3}\sin\left(\frac{t}{2}\right)}{z_{2}\sin\left(\frac{t}{2}\right)-z_{1}z_{3}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}&-\sin\left(\frac{t}{2}\right)\\\
0&\frac{z_{2}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}{z_{2}\sin\left(\frac{t}{2}\right)-z_{1}z_{3}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}&z_{1}\sin\left(\frac{t}{2}\right)\\\
0&0&z_{2}\sin\left(\frac{t}{2}\right)-z_{1}z_{3}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)\\\
\end{array}\right),$ $\displaystyle\beta_{6}$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{\cos\left(\frac{t}{2}\right)+iz_{3}\sin\left(\frac{t}{2}\right)}&-\frac{i}{\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)\cot\left(\frac{t}{2}\right)}\\\
0&0&\cos\left(\frac{t}{2}\right)+iz_{3}\sin\left(\frac{t}{2}\right)\\\
\end{array}\right),$ $\displaystyle\beta_{7}$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{z_{3}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)}&-\frac{1}{\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)\cot\left(\frac{t}{2}\right)}\\\
0&0&z_{3}\sin\left(\frac{t}{2}\right)+\cos\left(\frac{t}{2}\right)\\\
\end{array}\right),$ $\displaystyle\beta_{8}$
$\displaystyle=\left(\begin{array}[]{ccc}e^{-\frac{it}{2\sqrt{3}}}&0&0\\\
0&e^{-\frac{it}{2\sqrt{3}}}&0\\\ 0&0&e^{\frac{it}{\sqrt{3}}}\\\
\end{array}\right).$

In order to obtain the vector fields, we must compute the derivatives of
$A_{k}$ with respect to $t$ and evaluate in $t=0$. This reduces to determining
the derivatives of the coordinate functions

$\frac{d}{dt}f_{j,k}({z}_{j},t)$

in $t=0$ for all $j\in\\{1,2,3\\}$ and $k\in\\{1,\dots,8\\}$. The
corresponding vector fields are then, for $k\in\\{1,\dots,8\\}$, given by

$\displaystyle
X_{\lambda_{k}}=\left(\frac{d}{dt}\bigg{|}_{t=0}f_{1,k}({z}_{1},t)\right)\partial_{z_{1}}+\left(\frac{d}{dt}\bigg{|}_{t=0}f_{2,k}({z}_{2},t)\right)\partial_{z_{2}}+\left(\frac{d}{dt}\bigg{|}_{t=0}f_{3,k}({z}_{3},t)\right)\partial_{z_{3}}$

which yields the claim. ∎

Now we compute the explicit formula for the momentum map on
$\mathcal{O}^{\mathsf{SU}(3)}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})\simeq\mathsf{SL}(3,\mathbb{C})/B$
associated with vortex dynamics.

######  Theorem 4.3.

Let $K_{1}=1+|z_{1}|^{2}+|z_{2}|^{2}$ and
$K_{2}=1+|z_{3}|^{2}+|z_{1}z_{3}-z_{2}|^{2}$. The momentum map for the left
action of $\mathsf{SU}(3)$ on the generic coadjoint orbit
$\mathsf{SL}(3,\mathbb{C})/B$ is given by

$\displaystyle\mu:\mathsf{SL}(3,\mathbb{C})/B\to\mathfrak{su}(3)^{*},\qquad\begin{pmatrix}1&0&0\\\
z_{1}&1&0\\\ z_{2}&z_{3}&1\end{pmatrix}\mapsto(\mu_{ij})_{1\leq i,j\leq 3}$

where $(\mu_{ij})_{1\leq i,j\leq 3}$ is the traceless, anti-Hermitian matrix
with entries

$\displaystyle\mu_{11}$
$\displaystyle=\frac{1}{3}\left(\frac{x_{3}^{2}+y_{3}^{2}+2}{K_{2}}-\frac{x_{2}^{2}+y_{2}^{2}-1}{K_{1}}\right),$
$\displaystyle\mu_{22}$
$\displaystyle=\frac{1}{3}\left(-\frac{2x_{2}^{2}+2y_{2}^{2}+1}{K_{1}}-\frac{x_{3}^{2}+y_{3}^{2}-1}{K_{2}}\right),$
$\displaystyle\mu_{33}$ $\displaystyle=-(\mu_{11}+\mu_{22}),$
$\displaystyle\mu_{12}$
$\displaystyle=\frac{\left(iy_{1}-x_{1}\right)\left(x_{3}-iy_{3}\right)-iy_{2}+x_{2}}{K_{2}}-\frac{x_{1}-iy_{1}}{K_{1}},$
$\displaystyle\mu_{13}$
$\displaystyle=\frac{\left(iy_{1}-x_{1}\right)\left(x_{3}-iy_{3}\right)-iy_{2}+x_{2}}{K_{2}}-\frac{x_{1}-iy_{1}}{K_{1}},$
$\displaystyle\mu_{23}$
$\displaystyle=\frac{iy_{3}+x_{3}}{K_{2}}-\frac{\left(x_{1}+iy_{1}\right)\left(x_{2}-iy_{2}\right)}{K_{1}}.$

The remaining entries are determined by the fact that the matrix is anti-
Hermitian.

Let us prepare for the proof of Theorem 4.3 by computing the momentum map of
an easy example:

######  Example 4.4.

Let $J=\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$. Consider
the symplectic manifold $(\mathbb{R}^{2},\omega_{0}:=dx\wedge dy)$ and the
standard action of $\mathsf{SO}(2)\cong\mathsf{U}(1)\cong\mathbb{T}$ on
$\mathbb{R}^{2}$ by rotations, i.e., $\exp(\theta
J)=\left(\begin{smallmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{smallmatrix}\right)$ acts on
$\left(\begin{smallmatrix}x\\\ y\end{smallmatrix}\right)$ by

$\displaystyle\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}x\cos\theta-y\sin\theta\\\
x\sin\theta+y\cos\theta\end{pmatrix}.$

The vector field associated with the element $\exp(\theta J)$ is given by

$\displaystyle J_{M}=\frac{d}{d\theta}{\bigg{|}_{\theta=0}}\exp(\theta
J)\cdot\begin{pmatrix}x\\\ y\end{pmatrix}=y\partial_{x}-x\partial_{y}.$

The contraction of the symplectic form $\omega_{0}$ via the vector field
$J_{M}$ is

$\displaystyle\begin{pmatrix}y&-x\end{pmatrix}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}dx\\\ dy\end{pmatrix}=xdx+ydy.$

The map

$\mu:\mathbb{R}^{2}\to\mathbb{R},\quad(x,y)\mapsto\mu(x,y)=\frac{1}{2}(x^{2}+y^{2})$

is a moment map since it satisfies $\iota_{J_{M}}\omega=d\langle\mu,J\rangle$.

Recall that a $1$-form $\eta$ is called exact if there exists a function $F$
such that $dF=\eta$. The following example illustrate a technique how to find
such function $F$.

######  Example 4.5.

Let $dF=y^{2}e^{xy^{2}}dx+2xye^{xy^{2}}dy$ be a 1-form on $\mathbb{R}^{2}$.
Thus the (still unknown) function $F:\mathbb{R}^{2}\to\mathbb{R}$ satisfies
the following system of partial differential equations:

$\displaystyle\begin{aligned} \partial_{x}F(x,y)&=y^{2}e^{xy^{2}},\\\
\partial_{y}F(x,y)&=2xye^{xy^{2}}.\end{aligned}$

Integrating the first equation gives

$\displaystyle F(x,y)=\int y^{2}e^{xy^{2}}dx=e^{xy^{2}}+C(y),$

where $C(y)$ is a function depending on $y$. Using the second equation leads
to

$\displaystyle\partial_{y}F(x,y)=2xye^{xy^{2}}+C^{\prime}(y)=2xye^{xy^{2}}\Longleftrightarrow
C^{\prime}(y)=0.$

This means that $C\equiv c$ is constant and therefore the wanted function $F$
is given by $F(x,y)=e^{xy^{2}}+c$.

Now we are ready for

###### Proof of Theorem 4.3:.

By definition of the moment map, we must have

$\displaystyle d\langle\mu,\lambda_{k}\rangle=\iota_{X_{\lambda_{k}}}\omega,$

for all $\lambda_{k}\in\mathfrak{g}$ and induced vector fields
$X_{\lambda_{k}}$ from Lemma 4.2. Moreover, recall from (2.1.1) that the dual
pairing is given by the trace. Therefore we have

$\langle\mu,\lambda_{k}\rangle:\mathsf{SL}(3,\mathbb{C})/B\to\mathbb{R},\quad
x\mapsto\langle\mu(x),\lambda_{k}\rangle=\operatorname{\mathsf{trace}}(\mu(x)\lambda_{k}).$

The dual pairing explicitly becomes

$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&\frac{i}{2}&0\\\
\frac{i}{2}&0&0\\\ 0&0&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{i\mu_{12}}{2}+\frac{i\mu_{21}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&\frac{1}{2}&0\\\
-\frac{1}{2}&0&0\\\ 0&0&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{\mu_{21}}{2}-\frac{\mu_{12}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}\frac{i}{2}&0&0\\\
0&-\frac{i}{2}&0\\\ 0&0&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{i\mu_{11}}{2}-\frac{i\mu_{22}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&0&\frac{i}{2}\\\ 0&0&0\\\
\frac{i}{2}&0&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{i\mu_{13}}{2}+\frac{i\mu_{31}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&0&\frac{1}{2}\\\ 0&0&0\\\
-\frac{1}{2}&0&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{\mu_{31}}{2}-\frac{\mu_{13}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&\frac{i}{2}\\\
0&\frac{i}{2}&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{i\mu_{23}}{2}+\frac{i\mu_{32}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&\frac{1}{2}\\\
0&-\frac{1}{2}&0\\\ \end{array}\right)\right)$
$\displaystyle=\frac{\mu_{32}}{2}-\frac{\mu_{23}}{2},$
$\displaystyle\operatorname{\mathsf{trace}}\left(\left(\begin{array}[]{ccc}\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{21}&\mu_{22}&\mu_{23}\\\ \mu_{31}&\mu_{32}&\mu_{33}\\\
\end{array}\right)\left(\begin{array}[]{ccc}\frac{i}{2\sqrt{3}}&0&0\\\
0&\frac{i}{2\sqrt{3}}&0\\\ 0&0&-\frac{i}{\sqrt{3}}\\\
\end{array}\right)\right)$
$\displaystyle=\frac{i\left(\mu_{11}+\mu_{22}-2\mu_{33}\right)}{2\sqrt{3}}.$

In Lemma 3.17, we obtained the Hermitian metric $h=(h_{kl})_{1\leq k,l\leq n}$
on the flag manifold. Moreover, the (real) symplectic form is given by
$\omega=\frac{i}{2}\sum_{k,l=1}^{n}h_{kl}dz_{k}\wedge d\overline{z}_{l}$. In
terms of real coordinates
$(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})\in\mathbb{R}^{6}$ the matrix
representing the symplectic form is given by

$\displaystyle\omega:=\begin{pmatrix}\Im(h)&-\Re(h)\\\
\Re(h)&\Im(h)\end{pmatrix}$

with

$\displaystyle\Im(h)=\begin{pmatrix}0&\frac{x_{2}y_{1}-x_{1}y_{2}}{K_{1}^{2}}-\frac{y_{3}\left(x_{3}^{2}+y_{3}^{2}+1\right)}{K_{2}^{2}}&\frac{-\left(x_{3}\left(y_{1}-2x_{2}y_{3}\right)\right)-x_{3}^{2}y_{2}+y_{3}\left(x_{1}+y_{2}y_{3}\right)}{K_{2}^{2}}\\\
\frac{x_{1}y_{2}-x_{2}y_{1}}{K_{1}^{2}}+\frac{y_{3}\left(x_{3}^{2}+y_{3}^{2}+1\right)}{K_{2}^{2}}&0&\frac{x_{3}y_{2}-x_{2}y_{3}+y_{1}}{K_{2}^{2}}\\\
\frac{x_{3}^{2}y_{2}+x_{3}\left(y_{1}-2x_{2}y_{3}\right)-y_{3}\left(x_{1}+y_{2}y_{3}\right)}{K_{2}^{2}}&-\frac{x_{3}y_{2}-x_{2}y_{3}+y_{1}}{K_{2}^{2}}&0\\\
\end{pmatrix}$

and

$\displaystyle\Re(h)=\begin{pmatrix}\frac{x_{2}^{2}+y_{2}^{2}+1}{K_{1}^{2}}+\frac{x_{3}^{2}\left(2y_{3}^{2}+1\right)+x_{3}^{4}+y_{3}^{4}+y_{3}^{2}}{K_{2}^{2}}&-\frac{x_{1}x_{2}+y_{1}y_{2}}{K_{1}^{2}}-\frac{x_{3}\left(x_{3}^{2}+y_{3}^{2}+1\right)}{K_{2}^{2}}&\frac{y_{3}\left(2x_{3}y_{2}+y_{1}\right)+x_{2}\left(x_{3}^{2}-y_{3}^{2}\right)+x_{1}x_{3}}{K_{2}^{2}}\\\
-\frac{x_{1}x_{2}+y_{1}y_{2}}{K_{1}^{2}}-\frac{x_{3}\left(x_{3}^{2}+y_{3}^{2}+1\right)}{K_{2}^{2}}&\frac{x_{1}^{2}+y_{1}^{2}+1}{K_{1}^{2}}+\frac{x_{3}^{2}+y_{3}^{2}+1}{K_{2}^{2}}&-\frac{x_{1}+x_{2}x_{3}+y_{2}y_{3}}{K_{2}^{2}}\\\
\frac{y_{3}\left(2x_{3}y_{2}+y_{1}\right)+x_{2}\left(x_{3}^{2}-y_{3}^{2}\right)+x_{1}x_{3}}{K_{2}^{2}}&-\frac{x_{1}+x_{2}x_{3}+y_{2}y_{3}}{K_{2}^{2}}&\frac{K_{1}}{K_{2}^{2}}\\\
\end{pmatrix}.$

Evaluating $d\langle\mu,\lambda_{k}\rangle=\iota_{X_{\lambda_{k}}}\omega$
using the matrix representing $\omega$, we obtain the equations

$\displaystyle\begin{cases}d\left(\frac{i\mu_{12}}{2}+\frac{i\mu_{21}}{2}\right)&=\iota_{X_{\lambda_{1}}}\omega,\\\
d\left(\frac{\mu_{21}}{2}-\frac{\mu_{12}}{2}\right)&=\iota_{X_{\lambda_{2}}}\omega,\\\
d\left(\frac{i\mu_{13}}{2}+\frac{i\mu_{31}}{2}\right)&=\iota_{X_{\lambda_{4}}}\omega,\\\
d\left(\frac{\mu_{31}}{2}-\frac{\mu_{13}}{2}\right)&=\iota_{X_{\lambda_{5}}}\omega,\\\
d\left(\frac{i\mu_{23}}{2}+\frac{i\mu_{32}}{2}\right)&=\iota_{X_{\lambda_{6}}}\omega,\\\
d\left(\frac{\mu_{32}}{2}-\frac{\mu_{23}}{2}\right)&=\iota_{X_{\lambda_{7}}}\omega,\\\
d\left(\frac{i\mu_{11}}{2}-\frac{i\mu_{22}}{2}\right)&=\iota_{X_{\lambda_{3}}}\omega,\\\
d\left(\frac{i\left(\mu_{11}+\mu_{22}-2\mu_{33}\right)}{2\sqrt{3}}\right)&=\iota_{X_{\lambda_{8}}}\omega\end{cases}$
(4.3.1)

where we need to solve for the components $\mu_{ij}$ of the momentum map. The
1-forms on the left hand side are exact and therefore of the general form
$dF=\sum_{k=1}^{3}\frac{\partial F}{\partial
x_{k}}dx_{k}+\sum_{k=1}^{3}\frac{\partial F}{\partial y_{k}}dy_{k}$. The
1-forms on the right hand side are contractions and of the general form
$\sum_{k=1}^{3}G_{k}dx_{k}+\sum_{k=1}^{3}G_{3+k}dy_{k}$. Thus we must solve
$\frac{\partial F}{\partial x_{k}}=G_{k}$ and $\frac{\partial F}{\partial
y_{k}}=G_{k+3}$ for all $k\in\\{1,2,3\\}$. We now proceed as in Example 4.5.
Considering the coordinate $x_{1}$, we find

$\displaystyle F(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})=\int
G_{1}(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})dx_{1}+C(x_{2},x_{3},y_{1},y_{2},y_{3})$

and obtain therefore an expression for $F$. Using $\frac{\partial F}{\partial
x_{k}}=G_{2}$ we obtain

$\displaystyle G_{2}(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$
$\displaystyle=\frac{\partial F}{\partial
x_{2}}(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$
$\displaystyle=\frac{\partial}{\partial x_{2}}\int
G_{1}(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})dx_{1}+\frac{\partial}{\partial
x_{2}}C(x_{2},x_{3},y_{1},y_{2},y_{3})$

and therefore

$\frac{\partial}{\partial x_{2}}C=G_{2}-\frac{\partial}{\partial x_{2}}\int
G_{1}$

which yields

$C=\int\left(G_{2}-\frac{\partial}{\partial x_{2}}\int
G_{1}\right)dx_{2}+C^{\prime}(x_{3},y_{1},y_{2},y_{3}).$

Iterating this procedure, we determine $F$ in terms of $G_{1},\dots,G_{6}$.
Therefore, if $G_{1},\dots,G_{6}$ are explicitly given, we can find an
explicit formula for $F$.

Now we will apply this procedure to the systems of equations given in (4.3.1).
We start with the coupled system

$\displaystyle\begin{cases}d\left(\frac{i\mu_{12}}{2}+\frac{i\mu_{21}}{2}\right)&=\iota_{X_{\lambda_{1}}}\Omega,\\\
d\left(\frac{\mu_{21}}{2}-\frac{\mu_{12}}{2}\right)&=\iota_{X_{\lambda_{2}}}\Omega.\end{cases}$

By integration we obtain the following expressions. Note that, in our
situation, the function $C$ from above is constant and may be chosen to be
zero.

$\displaystyle\frac{i}{2}\left(\mu_{12}+\mu_{21}\right)$
$\displaystyle=\frac{1}{2}\left(-\frac{x_{1}\left(x_{3}^{2}+y_{3}^{2}\right)-x_{2}x_{3}-y_{2}y_{3}}{K_{2}}-\frac{x_{1}}{K_{1}}\right)=:\alpha,$
$\displaystyle\frac{1}{2}\left(\mu_{21}-\mu_{12}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{x_{3}y_{2}-x_{3}^{2}y_{1}-y_{3}\left(x_{2}+y_{1}y_{3}\right)}{K_{2}}-\frac{y_{1}}{K_{1}}\right)=:\beta.$

We now can solve for $\mu_{21}$ via

$\displaystyle i\mu_{21}=\alpha+i\beta$
$\displaystyle=\frac{1}{2}\left(\frac{-\left(x_{1}+iy_{1}\right)\left(x_{3}+iy_{3}\right)+x_{2}+iy_{2}}{K_{2}}-\frac{x_{1}+iy_{1}}{K_{1}}\right)$
$\displaystyle=\frac{1}{2}\left(-\frac{z_{1}}{K_{1}}+\frac{-z_{1}z_{3}+z_{2}}{K_{2}}\right)$

and find

$\displaystyle{\mu_{21}=-\frac{i}{2}\left(-\frac{z_{1}}{K_{1}}+\frac{-z_{1}z_{3}+z_{2}}{K_{2}}\right)}.$
(4.3.2)

The second pair of coupled equations is

$\displaystyle\begin{cases}d\left(\frac{i\mu_{13}}{2}+\frac{i\mu_{31}}{2}\right)&=\iota_{X_{\lambda_{4}}}\Omega,\\\
d\left(\frac{\mu_{31}}{2}-\frac{\mu_{13}}{2}\right)&=\iota_{X_{\lambda_{5}}}\Omega\end{cases}$

which can be integrated as

$\displaystyle\frac{i\mu_{13}}{2}+\frac{i\mu_{31}}{2}$
$\displaystyle=\frac{1}{2}\left(\frac{-x_{2}+x_{1}x_{3}-y_{1}y_{3}}{K_{2}}-\frac{x_{2}}{K_{1}}\right)=:\gamma,$
$\displaystyle\frac{\mu_{31}}{2}-\frac{\mu_{13}}{2}$
$\displaystyle=\frac{1}{2}\left(\frac{x_{3}y_{1}+x_{1}y_{3}-y_{2}}{K_{2}}-\frac{y_{2}}{K_{1}}\right)=:\delta$

We optain

$\displaystyle
i\mu_{31}=\gamma+i\delta=-\frac{x_{2}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}i}y_{2}}{2K_{1}}-\frac{-x_{3}y_{1}-x_{1}\left(x_{3}+y_{3}\right)+x_{2}+y_{2}+y_{1}y_{3}}{2K_{2}}$

and finally

$\displaystyle\mu_{31}=\frac{-i}{2}\left(-\frac{x_{2}+iy_{2}}{K_{1}}-\frac{-\left(x_{1}+iy_{1}\right)\left(x_{3}+iy_{3}\right)+x_{2}+iy_{2}}{K_{2}}\right).$

The next pair of equations is

$\displaystyle\begin{cases}d\left(\frac{i\mu_{23}}{2}+\frac{i\mu_{32}}{2}\right)&=\iota_{X_{\lambda_{6}}}\Omega,\\\
d\left(\frac{\mu_{32}}{2}-\frac{\mu_{23}}{2}\right)&=\iota_{X_{\lambda_{7}}}\Omega.\end{cases}$

We compute

$\displaystyle\frac{i\mu_{23}}{2}+\frac{i\mu_{32}}{2}$
$\displaystyle=\frac{1}{2}\left(\frac{x_{3}}{K_{2}}-\frac{x_{1}x_{2}+y_{1}y_{2}}{K_{1}}\right)=:\zeta,$
$\displaystyle\frac{\mu_{32}}{2}-\frac{\mu_{23}}{2}$
$\displaystyle=\frac{1}{2}\left(\frac{x_{2}y_{1}-x_{1}y_{2}}{K_{1}}-\frac{y_{3}}{K_{2}}\right)=:\eta$

and obtain $\zeta+i\eta=i\mu_{32}$ and therefore

$\displaystyle i\mu_{32}$
$\displaystyle=\frac{1}{2}\left(\frac{x_{3}}{K_{2}}-\frac{x_{1}x_{2}+y_{1}y_{2}}{K_{1}}\right)+i\frac{1}{2}\left(\frac{x_{2}y_{1}-x_{1}y_{2}}{K_{1}}-\frac{y_{3}}{K_{2}}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{-x_{1}x_{2}-y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{K_{1}}+\frac{x_{3}-iy_{3}}{K_{2}}\right).$

In order to determine $\mu_{33}$, we integrate the last equation in (4.3.1)
and obtain

$\displaystyle\frac{i\left(\mu_{11}+\mu_{22}-2\mu_{33}\right)}{2\sqrt{3}}=\frac{\sqrt{3}}{4}\left(\frac{x_{2}^{2}+y_{2}^{2}}{K_{1}}-\frac{1}{K_{2}}\right).$

Using the fact that the matrix $M$ is traceless, i.e.
$\mu_{11}+\mu_{22}+\mu_{33}=0$, the left-hand side reduces to

$\displaystyle\frac{-3i\mu_{33}}{2\sqrt{3}}=\frac{\sqrt{3}}{4}\left(\frac{x_{2}^{2}+y_{2}^{2}}{K_{1}}-\frac{1}{K_{2}}\right)$

and therefore

$\displaystyle\mu_{33}=\frac{i}{2}\left(\frac{x_{2}^{2}+y_{2}^{2}}{K_{1}}-\frac{1}{K_{2}}\right).$

On the other hand, using $2(\mu_{11}+\mu_{22})=-2\mu_{33}$, we have

$\displaystyle\frac{i}{2\sqrt{3}}\left(\mu_{11}+\mu_{22}-2\mu_{33}\right)=\frac{3i}{2\sqrt{3}}\left(\mu_{11}+\mu_{22}\right)=\frac{\sqrt{3}}{4}\left(\frac{x_{2}^{2}+y_{2}^{2}}{K_{1}}-\frac{1}{K_{2}}\right).$

Moreover, integrating the remaining Cartan equation
$d\left(\frac{i\mu_{11}}{2}-\frac{i\mu_{22}}{2}\right)=\iota_{X_{\lambda_{3}}}\omega$
gives

$\frac{i}{2}(\mu_{11}-\mu_{22})=\kappa.$

This yields the system of equations

$\displaystyle\begin{cases}\frac{3i}{2\sqrt{3}}\left(\mu_{11}+\mu_{22}\right)&=\lambda,\\\
\frac{i}{2}\left({\mu_{11}}-{\mu_{22}}\right)&=\kappa\end{cases}\quad\Longleftrightarrow\quad\begin{cases}\frac{3i}{2\sqrt{3}}\left(\mu_{11}+\mu_{22}\right)&=\lambda,\\\
\frac{3i}{2\sqrt{3}}\left({\mu_{11}}-{\mu_{22}}\right)&=\frac{3}{\sqrt{3}}\kappa.\end{cases}$

Adding (resp. subtracting) the equations gives

$\displaystyle\mu_{11}$
$\displaystyle=-\frac{\sqrt{3}}{3}i\left(\alpha+\frac{3}{\sqrt{3}}\beta\right)=-\frac{i}{\sqrt{3}}\left(\frac{x_{2}^{2}+y_{2}^{2}-1}{2\sqrt{3}K_{1}}-\frac{x_{3}^{2}+y_{3}^{2}+2}{2\sqrt{3}K_{2}}\right),$
$\displaystyle\mu_{22}$
$\displaystyle=-\frac{\sqrt{3}}{3}i\left(\alpha-\frac{3}{\sqrt{3}}\beta\right)=-\frac{i}{\sqrt{3}}\left(\frac{2x_{2}^{2}+2y_{2}^{2}+1}{2\sqrt{3}K_{1}}+\frac{x_{3}^{2}+y_{3}^{2}-1}{2\sqrt{3}K_{2}}\right),$
$\displaystyle\mu_{33}$ $\displaystyle=-(\mu_{11}+\mu_{22}).$

Now as we are working over $\mathfrak{su}^{*}(3)$ we have obtained all the
entries of the matrix. ∎

##  5\. Green’s function and the vortex Hamiltonian for $\mathbb{CP}^{n}$

In Section 4, we recalled the point vortex momentum map on $\mathbb{CP}^{2}$
and computed the one on $\mathbb{F}_{1,2}(\mathbb{C}^{3})$, i.e., on the
degenerate and generic orbits of the action of $\mathsf{SU}(3)$. The aim of
this section is to determine the Hamiltonian for the point vortex problem on
$\mathbb{CP}$. An important ingredient here is the fundamental solution of the
Laplace-Beltrami operator.

###  5.1. The Laplace-Beltrami operator

Let $(M,g)$ be a Riemannian manifold of dimension $n$ with local coordinates
$x=(x_{1},\dots,x_{n})$ in which we express the metric by the symmetric
$(n\times n)$-matrix $(g_{ij}):=(g_{ij})_{1\leq i,j\leq n}$. The inverse of
this matrix is denoted by $(g_{ij})^{-1}=:(g^{ij})$. The Riemannian volume
form on $(M,g)$ is given in local coordinates by

$\sqrt{|\det(g)|}dx_{1}\wedge\cdots\wedge dx_{n}=:d\mu.$

Let $f:M\to\mathbb{R}$ a smooth function. Then the Laplace-Beltrami operator
$\Delta$ on $M$ is given in local coordinates $(x_{1},\dots,x_{n})$ by

$\displaystyle\Delta
f=-\frac{1}{\sqrt{|\det(g_{ij})|}}\sum_{i,j=1}^{n}\partial_{x_{j}}g^{ij}\sqrt{|\det(g_{ij})|}\
\partial_{x_{i}}f.$

which, on $\mathbb{R}^{n}$ equipped with the Euclidean metric, yields
$\Delta=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$. The big diagonal
$\operatorname{\mathsf{Diag}}_{N}(M)$ of the $N$-fold product
$M\times\dots\times M$ is defined as

$\operatorname{\mathsf{Diag}}_{N}(M):=\\{(y_{1},\dots,y_{N})\in M^{N}\mid
y_{j}=y_{k}\mbox{ for some }j\neq k\mbox{ with }1\leq j,k\leq N\\},$

i.e., it consists of all $N$-tuple point on $M$ for which at least two points
coincide. If $(M,g)$ is compact, then, for the Laplace-Beltrami operator,
there exists a function $G:(M\times
M)\setminus{\operatorname{\mathsf{Diag}}_{2}(M)}\to\mathbb{R}$, referred to as
Green’s function, satisfying the following properties (see for instance
Section 2.3 in Aubin [Aub98]):

  1. (1)

For all functions $\phi\in C^{2}(M,\mathbb{R})$, we have

$\displaystyle\phi(p)=\frac{1}{\operatorname{\mathsf{vol}}(M)}\int_{M}\phi(q)\
d\mu(q)+\int_{M}G(p,q)\Delta\phi(q)\ d\mu(q).$ (5.1.1)

  2. (2)

$G$ is smooth on $(M\times M)\setminus\operatorname{\mathsf{Diag}}_{2}(M)$.

  3. (3)

$G$ is symmetric, i.e. $G(p,q)=G(q,p)$ for all $p,q\in M$.

  4. (4)

We have $\int_{M}G(p,q)\ d\mu(q)=\text{constant}$ for all $p\in M$.

Green’s function is also called a fundamental solution of the Laplace-Beltrami
operator.

######  Examples 5.1.

  1. (1)

Green’s function on $\mathbb{R}^{2}$ equipped with the Euclidean metric is
given by

$G(x,y)=-\frac{1}{2\pi}\ln|x-y|$

where $|x-y|:=\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}}$ is the Euclidean
distance. Note that $|x-y|$ coincides with the geodesic distance $r(x,y)$ of
the Euclidean metric.

  2. (2)

Consider the unit sphere $\mathbb{S}^{2}$ with $x=(\sin\theta\cos\phi,\
\sin\theta\sin\phi,\ \cos\theta)\in\mathbb{S}^{2}$ and
$y=(\sin\theta^{\prime}\cos\phi^{\prime},\
\sin\theta^{\prime}\sin\phi^{\prime},\ \cos\theta^{\prime})\in\mathbb{S}^{2}$.
Then Green’s function $G(x,y)$ for the spherical Laplace operator is given by
(see Dritschel [Dri88])

$G(x,y)=\frac{1}{2\pi}\log(1-\cos\Theta),$

where
$\Theta=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime})$.

On the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$, we have

######  Proposition 5.2.

The Laplace operator on the flag manifold is given by

$\Delta_{\mathbb{F}_{1,2}(\mathbb{C}^{3})}=\Delta_{\mathbb{CP}^{2}}+\Delta_{R}$

where

$\displaystyle\Delta_{\mathbb{CP}^{2}}=\sum_{j,k=1}^{2}(1+\delta_{jk}{z_{k}}{\overline{z}_{j}})\partial_{z_{j}}\partial_{{\overline{z}}_{k}}$

is the Laplace operator on $\mathbb{CP}^{2}$ and $\Delta_{R}$ a correction
term given by

$\displaystyle\Delta_{R}$
$\displaystyle=\frac{K_{1}^{2}}{K_{2}}\left(\partial_{z_{1}}\partial_{\overline{z}_{1}}+z_{3}\partial_{z_{1}}\partial_{\overline{z}_{2}}+\overline{z}_{3}\partial_{z_{2}}\partial_{\overline{z}_{1}}+|z_{3}|^{2}\partial_{z_{2}}\partial_{\overline{z}_{2}}\right)+\left(K_{1}(1+|z_{3}|^{2})+\frac{K_{2}^{2}}{K_{1}}\right)\partial_{z_{3}}\partial_{\overline{z}_{3}}$
$\displaystyle\quad+(\overline{z}_{1}+z_{3}\overline{z}_{2})(\overline{z}_{1}z_{2}-z_{3}-z_{3}|z_{1}|^{2})\partial_{z_{1}}\partial_{\overline{z}_{3}}+(\overline{z}_{1}+\overline{z}_{2}z_{3})\left((1+|z_{2}|^{2})-z_{1}\overline{z}_{2}z_{3}\right)\partial_{z_{2}}\partial_{\overline{z}_{3}}$
$\displaystyle\quad+(z_{1}+\overline{z}_{3}z_{2})(z_{1}\overline{z}_{2}-\overline{z}_{3}-z_{3}|z_{1}|^{2})\partial_{z_{3}}\partial_{\overline{z}_{1}}+({z}_{1}+{z}_{2}\overline{z}_{3})\left((1+|z_{2}|^{2})-{\overline{z}}_{1}{z}_{2}\overline{z}_{3}\right)\partial_{z_{3}}\partial_{\overline{z}_{2}}.$

where $K_{1}$ and $K_{2}$ are the functions from Lemma 3.15.

###### Proof.

On an $n$-dimensional Kähler manifold $(M,h)$ with Kähler potential $K_{M}$,
the Laplace operator $\Delta$ is given by

$\displaystyle\Delta=2\sum_{i,j=1}^{n}h^{ij}\partial_{i}\overline{\partial}_{j}K_{{M}}.$

Using the expression for the Kähler potential on the flag manifold from Lemma
3.15, one obtains the expression in the proposition by direct calculations. ∎

###  5.2. The Hamiltonian for point vortex dynamics

Let $(M,\omega)$ be a symplectic manifold and $N\in\mathbb{N}$. For $1\leq
k\leq N$, let $\tau_{k}:\Pi_{k=1}^{N}M_{k}\to M$ be the projection on the
$k$th factor and let $\Gamma_{1},\dots,\Gamma_{N}\in\mathbb{R}^{\neq 0}$.
Consider the space
$\mathcal{M}:=\Pi_{k=1}^{N}M\setminus\operatorname{\mathsf{Diag}}_{N}(M)$ and
endow it with the symplectic form
$\Omega:=\Omega(\Gamma):=\sum_{k=1}^{N}\Gamma_{k}\tau_{k}^{*}\omega$. Recall
Green’s function $G:(M\times
M)\setminus\operatorname{\mathsf{Diag}}_{N}(M)\to\mathbb{R}$ defined by the
expression (5.1.1) and define the so-called Robin function (see also Dritschel
$\&$ Boatto [DB15])

$\displaystyle R:M\to\mathbb{R},\quad R(t):=\lim_{\tilde{t}\to
t}\left(G(\tilde{t},t)-\frac{1}{2\pi}\log d(\tilde{t},t)\right).$

The Hamiltonian $H:\mathcal{M}\to\mathbb{R}$,

$\displaystyle H(s_{1},\dots,s_{N}):=\sum_{1\leq i<j\leq N}
\Gamma_{i}\Gamma_{j}G(s_{i},s_{j})+\sum_{k=1}^{N}\Gamma_{k}^{2}R_{g}(s_{k})$
(5.2.1)

describes the dynamics of $N$ vortices with vortex strength
$\Gamma_{k}\in\mathbb{R}^{\neq 0}$ for $k=1,\dots,N$ on the phase space
$\mathcal{M}$. Green’s function $G$ describes the interaction between pairs of
distinct vortices and the Robin function takes self-interactions into account
(for more details, see Lin [Lin41]).

To study the vortex dynamics on an explicitly given symplectic manifold, we
need explicit formulas for Green’s function. Unfortunately, Green’s functions
are explicitly known only for certain classes of manifolds as for instance
planes, hyperbolic planes, and $2$-spheres, see Galajinsky [Gal22], Lim $\&$
Montaldi $\&$ Roberts [LMR01], Montaldi $\&$ Nava-Gaxiola [MNG14].

Motivated by the study of the generic and degenerate coadjoint orbits of
$\mathsf{SU}(3)$, we naturally are interested in the Green’s functions on
these coadjoint orbits. The degenerate coadjoint orbit is $\mathbb{CP}^{2}$
and, in the case of $\mathsf{SU}(n+1)$, one of the degenerate coadjoint orbits
is $\mathbb{CP}^{n}$.

###  5.3. Green’s function and the Hamiltonian on the coadjoint orbit
$\mathbb{CP}^{n}$

The aim of this subsection is to obtain an explicit formula for Green’s
function on $\mathbb{CP}^{n}$. This will allow us then to write down the
Hamiltonian for the point vortex dynamics on $\mathbb{CP}^{n}$ explicitly.

Given an arbitrary compact Riemannian manifold, the explicit computation of
Green’s function is not obvious. However, there are certain homogeneous spaces
for which methods are available to obtain an explicit formula for Green’s
function. Among these spaces, there are compact rank one symmetric spaces,
briefly CROSS spaces. All CROSS spaces are given by the following list: the
$n$-sphere $\mathbb{S}^{n}$, the projective spaces $\mathbb{KP}^{n}$ for
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C},\mathbb{H}\\}$, and the octonionic
plane $\mathbb{OP}^{2}$.

CROSS spaces are special examples of so-called locally harmonic Blaschke
manifolds (see Besse [Bes78]) for which the following result was established.

######  Proposition 5.3 (Beltrán & Corral & Criado del Rey [BCdR19]).

Let $M$ be a locally harmonic Blaschke manifold and denote the geodesic
distance between two points $x,y\in M$ by $r(x,y)=:r$. Then, Green’s function
on $M$ is given by $G(x,y)=\varphi(r)$ where $\varphi$ is determined by the
differential equation

$\displaystyle\varphi^{\prime}(r)=-\frac{1}{r^{n-1}V_{M}(r)\operatorname{\mathsf{vol}}(M)}\int_{r}^{\operatorname{inj}(M)}t^{n-1}V_{M}(t)dt$
(5.3.1)

where $V_{M}$ is the so-called volume density and $\operatorname{inj}(M)$ is
the injectivity radius of the manifold.

Recall that the diameter of a Riemannian manifold $(M,g)$ is defined by

$\operatorname{\mathsf{diam}}(M):=\sup_{x,y\in M}r(x,y)$

where $r(x,y)$ is the geodesic distance between $x,y\in(M,g)$. We now focus on
the CROSS space $M=\mathbb{CP}^{n}$ where, in fact, the injectivity radius is
equal to the diameter. The density function $V_{\mathbb{CP}^{n}}$ was
explicitly determined by Kreyssig [Kre10] as

$\displaystyle
V_{\mathbb{CP}^{n}}(r)=\frac{2^{2n-1}}{r^{n-1}}\sin^{2n-1}(r)\cos(r).$ (5.3.2)

In order to solve the differential equation (5.3.1) for $M=\mathbb{CP}^{n}$ we
need the following technical result.

######  Lemma 5.4.

Let $n\in\mathbb{N}^{>0}$. Then

$\displaystyle\int\frac{1-\sin^{2n}(x)}{\sin^{2n-1}(x)\cos(x)}\
dx=\log(\sin(x))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}(x)}.$

###### Proof.

We start with

$\displaystyle\int\frac{1-\sin^{2n}(x)}{\sin^{2n-1}(x)\cos(x)}\ dx$
$\displaystyle=\int\frac{1}{\sin^{2n-1}(x)\cos(x)}\ dx-\int\tan(x)\ dx$
$\displaystyle=\int\frac{1}{\sin^{2n-1}(x)\cos(x)}\ dx+\log(\cos(x)).$

Using $1+\cot^{2}(x)=\csc^{2}(x)$, we obtain

$\displaystyle\int\frac{1}{\sin^{2n-1}(x)\cos(x)}\ dx$
$\displaystyle=\int\sec(x)\csc^{2n-1}(x)\ dx=\int\sec(x)\csc^{2n-2}(x)\csc(x)\
dx$ $\displaystyle=\int\sec(x)\csc(x)(1+\cot^{2}(x))^{n-1}\ dx.$

Using the binomial formula

$\displaystyle(1+\cot^{2}(x))^{n-1}=\sum_{k=0}^{n-1}{n-1\choose
k}\cot^{2k}(x)$

we obtain

$\displaystyle\int\sec(x)\csc(x)\sum_{k=0}^{n-1}{n-1\choose k}\cot^{2k}(x)\
dx=\sum_{k=0}^{n-1}{n-1\choose k}\int\sec(x)\csc(x)\cot^{2k}(x)\ dx.$

Since

$\displaystyle\sec(x)\csc(x)\cot^{\ell}(x)=\frac{1}{\sin(x)\cos(x)}\frac{\cos^{\ell}(x)}{\sin^{\ell}(x)}=\frac{\cos^{\ell-1}(x)}{\sin^{\ell+1}(x)}=\frac{\cot^{\ell-1}(x)}{\sin^{2}(x)}$

the integral becomes

$\displaystyle\int\cot^{\ell-1}\csc^{2}(x)\ dx.$

By using the substitution $v=\cot(x)$ and $dv=-\csc^{2}(x)dx$ we have

$-\int v^{\ell-1}\ dv=-\frac{v^{\ell}}{\ell}+C=-\frac{\cot^{\ell}(x)}{\ell}+C$

where $C$ is the integration constant. Now, for $\ell=2k$, we can rewrite

$\sum_{k=0}^{n-1}{n-1\choose k}\int\sec(x)\csc(x)\cot^{2k}(x)\
dx=\int\sec(x)\csc(x)\ dx-\sum_{k=1}^{n-1}{n-1\choose
k}\frac{\cot^{2k}(x)}{2k}.$

We compute

$\int\sec(x)\csc(x)dx=\log(\sin(x))-\log(\cos(x)).$

This means for our original integral

$\displaystyle\int\frac{1-\sin^{2n}(x)}{\sin^{2n-1}(x)\cos(x)}dx$
$\displaystyle\quad=\log(\sin(x))-\log(\cos(x))-\sum_{k=1}^{n-1}{n-1\choose
k}\frac{\cot^{2k}(x)}{2k}+\log(\cos(x))$
$\displaystyle\quad=\log(\sin(x))-\sum_{k=1}^{n-1}{n-1\choose
k}\frac{\cot^{2k}(x)}{2k}.$

By using repeatedly

$\cot^{2}(x)=\frac{1}{\sin^{2}(x)},$

we finally obtain

$\int\frac{1-\sin^{2n}(x)}{\sin^{2n-1}(x)\cos(x)}dx=\log(\sin(x))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}(x)}.$

∎

We now obtain

######  Theorem 5.5.

Green’s function on $\mathbb{CP}^{n}$ with the Fubini-Study metric is given by
$G:\mathbb{CP}^{n}\times\mathbb{CP}^{n}\setminus\operatorname{\mathsf{Diag}}_{2}(\mathbb{CP}^{n})\to\mathbb{R}$
with

$\displaystyle
G(\xi,\eta)=-\frac{1}{2n\cdot\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})}\left(\log(\sin(r(\xi,\eta)))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}(r(\xi,\eta))}\right)$

where
$r(\xi,\eta)=\arccos\sqrt{\frac{\langle\xi,\eta\rangle\langle\eta,\xi\rangle_{H}}{\langle\xi,\xi\rangle\langle\eta,\eta\rangle_{H}}}$
is the geodesic distance between the two point in $\mathbb{CP}^{n}$ and
$\langle\cdot,\cdot\rangle_{H}$ is the Hermitian inner product.

###### Proof.

Recall from Equation (5.3.2) that the volume density of $\mathbb{CP}^{n}$ is

$\displaystyle{}V_{\mathbb{CP}^{n}}(r)$
$\displaystyle=\frac{2^{2n-1}\sin^{2n-1}(r)\cos(r)}{r^{n-1}}$

and that

$\operatorname{inj}(\mathbb{CP}^{n})=\operatorname{\mathsf{diam}}(\mathbb{CP}^{n})=\frac{\pi}{2}\quad\text{
and }\quad\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})=\frac{\pi^{n}}{n!}.$

Therefore the ODE for $\varphi(r)$ from (5.3.1) can be written as

$\varphi^{\prime}(r)=-\frac{1}{r^{n-1}V_{\mathbb{CP}^{n}}(r)\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})}\int_{r}^{\operatorname{\mathsf{diam}}(\mathbb{CP}^{n})}t^{n-1}V_{\mathbb{CP}^{n}}(t)\
dt$

and gives rise to the following equation:

$\displaystyle\varphi^{\prime}(r)$
$\displaystyle=-\frac{1}{\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})\sin^{2n-1}(r)\cos(r)}\int_{r}^{\frac{\pi}{2}}\sin^{2n-1}(t)\cos(t)dt$
$\displaystyle=-\frac{1}{\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})\sin^{2n-1}(r)\cos(r)}\
\frac{1}{2n}(1-\sin^{2n}(r)).$

Solving this ODE gives the following formula for the fundamental solution

$\displaystyle\varphi(r)$
$\displaystyle=-\frac{1}{2n\cdot\operatorname{\mathsf{vol}}(\mathbb{CP}^{n})}\int\frac{1-\sin^{2n}(r)}{\sin^{2n-1}(r)\cos(r)}\
dr.$

Using Lemma 5.4 now gives the result. ∎

######  Theorem 5.6.

The Hamiltonian for the $N$ point vortex dynamics on the projective space
$\mathbb{CP}^{n}$ is explicitly given by

$\displaystyle
H:\bigl{(}\mathbb{CP}^{n}\bigr{)}^{N}\setminus\operatorname{\mathsf{Diag}}_{N}(\mathbb{CP}^{n})\to\mathbb{R},$
$\displaystyle
H(\zeta)=-\frac{1}{2(n-1)!\pi^{n}}\sum_{\alpha<\beta}^{N}\Gamma_{\alpha}\Gamma_{\beta}\left(\log(\sin(r(\zeta_{\alpha},\zeta_{\beta})))-\sum_{j=1}^{n-1}\frac{1}{2j\sin^{2j}({r(\zeta_{\alpha},\zeta_{\beta})})}\right)$

where $\zeta=(\zeta_{1},\dots,\zeta_{N})$ and
$r(\zeta_{\alpha},\zeta_{\beta})$ is the geodesic distance on
$\mathbb{CP}^{n}$ between the two points given by

$\displaystyle
r(\zeta_{\alpha},\zeta_{\beta})=\arccos\sqrt{\frac{\langle\zeta_{\alpha},\zeta_{\beta}\rangle_{H}\langle\zeta_{\beta},\zeta_{\alpha}\rangle_{H}}{\langle\zeta_{\alpha},\zeta_{\alpha}\rangle_{H}\langle\zeta_{\beta},\zeta_{\beta}\rangle_{H}}},$

where $\langle\cdot,\cdot\rangle_{H}$ is the Hermitian inner product.

###### Proof.

Equation (5.2.1) gives the formal expression for the Hamiltonian of the N
point vortex problem on a manifold. Using the explicit expression for the
Green function on $\mathbb{CP}^{n}$ from Theorem 5.5 yields the result. ∎

The Hamiltonian vector field $\mathcal{X}_{H}$ (and then also the equations of
motion) for the Hamiltonian from Theorem 5.6 can be computed either by using
the implicit formula involving the symplectic form or by making use of an
compatible almost complex structure $J$ for the Fubini-Study metric, i.e., via
$X^{H}=J\operatorname{grad}(H)$.

###  5.4. The Hamiltonian on the flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^{3})$

As we saw in previous subsections, the explicit knowledge of Green’s function
is quite rare which makes it complicated to obtain an explicit expression for
the Hamiltonian of the point vortex problem in many situations.

We do not yet have an explicit formula for Green’s function and the Laplacian
on the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^{3})$ so that we also do not
yet have an explicit expression for the Hamiltonian of the point vortex
problem. One idea to approach this open question may be the fibration

$\mathbb{S}^{2}\longrightarrow
W^{6}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})\longrightarrow\mathbb{C}\mathbb{P}^{2}$

together with the hope to deduce Green’s function on
$W^{6}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})$ from those on $\mathbb{S}^{2}$
and $\mathbb{C}\mathbb{P}^{2}$ and thus obtain the Hamiltonian on
$W^{6}\simeq\mathbb{F}_{1,2}(\mathbb{C}^{3})$.

In fact, this poses the more general question of the behaviour of Green’s
function with respect to fibrations in general. But this is beyond the scope
of the present paper.

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