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\title{\textsf{New BPS Solitons in 2+1 Dimensional Noncommutative }
$CP^{1}$\textsf{Model}}
\author{
\textsf{Hideharu Otsu }
\thanks{otsu@vega.aichi-u.ac.jp}\\
Faculty of Economics, \\
Aichi University, Toyohashi, Aichi 441-8522, Japan
\and
\textsf{Toshiro Sato}
\thanks{tsato@matsusaka-u.ac.jp}\\
Faculty of Policy Science, Matsusaka University, \\
Matsusaka, Mie 515-8511, Japan
\and
\textsf{Hitoshi Ikemori }
\thanks{ikemori@biwako.shiga-u.ac.jp}\\
Faculty of Economics, Shiga University, \\
Hikone, Shiga 522-8522, Japan
\and
\textsf{Shinsaku Kitakado}\\
Department of Information Sciences, Meijo University, \\
Tempaku, Nagoya 486-8502, Japan
}
\date{}
\maketitle

\begin{abstract}
Investigating the solitons in the non-commutative $CP^{1}$ model, we have
found a new set of BPS solitons which does not have counterparts in the
commutative model.

\end{abstract}

\newpage

\section[ Introduction]{Introduction}

\smallskip Field theories on the non-commutative space have been extensively
studied in the last few years. Particularly, BPS solitons are interesting,
because they might not share the common features with those on the
commutative
space\cite{Harvey:2001yn}\cite{Nekrasov:1998ss}\cite{Gopakumar:2000zd}.

Solitons in the non-commutative $CP^{1}$ model have been studied in
\cite{Lee:2000ey} and further developed in \cite{Furuta:2002ty} in
connection
with the dynamical aspects of the theory. The non-BPS solitons, that do not
exist in the commutative case, have been studied in \cite{Furuta:2002nv}.
These investigations were reviewed in \cite{Murugan:2002rz}.

In this paper, we report on a set of new BPS solitons in the non-commutative
$CP^{1}$ model, that does not exist in the commutative limit.

We consider the $CP^{1\text{ }}$model on 2+1 dimensional non-commutative
spacetime. The space coordinates obey the commutation relation%

\begin{equation}
\left[ x,y\right] =i\theta,
\end{equation}
or%

\begin{equation}
\left[ z,\bar{z}\right] =\theta>0,
\end{equation}
in terms of the complex variables, $z=\frac{1}{\sqrt{2}}(x+iy)$ and $\bar
{z}=\frac{1}{\sqrt{2}}(x-iy)$. The Hilbert space can be described in
terms of
the energy eigenstates $\left\vert n\right\rangle $ of the harmonic
oscillator
whose creation and annihilation operators are $\bar{z}$ and $z$
respectively,%

\begin{align}
z\left| n\right\rangle & =\sqrt{\theta n}\left| n-1\right\rangle ,\ \ \\
\bar{z}\left| n\right\rangle & =\sqrt{\theta(n+1)}\left| n+1\right\rangle
,\nonumber
\end{align}
The $CP^{1}$ lagrangian is%

\begin{equation}
L=\text{$\mathrm{Tr}$}(\left\vert D_{t}\Phi\right\vert ^{2}-\left\vert
D_{z}\Phi\right\vert ^{2}-\left\vert D_{\bar{z}}\Phi\right\vert ^{2}),
\end{equation}
where $\Phi$ is a 2-component complex vector with the constraint
$\Phi^{\dagger}\Phi=1$. We consider $\Phi$ to be the fundamental field and
thus to be non-singular. Tr denotes the trace over the Hilbert space
as\smallskip%
\begin{equation}
\text{$\mathrm{Tr}$}\mathcal{O}=2\pi\theta\sum_{n=0}^{\infty}\left\langle
n\right\vert \mathcal{O}\left\vert n\right\rangle .
\end{equation}
The covariant derivative is defined by
\begin{equation}
D_{a}\Phi=\partial_{a}\Phi-i\Phi A_{a},\quad A_{a}=-i\Phi^{\dagger}%
\partial_{a}\Phi,\quad\left( a=t,z,\bar{z}\right) ,
\end{equation}
where $\partial_{z}=-\theta^{-1}\left[ \bar{z},\ \right] $ and
$\partial_{\bar{z}}=\theta^{-1}\left[ z,\ \right] $.

For the static configurations, topological charge and static energy are
given
by
\begin{equation}
Q=\frac{1}{2\pi}\mathrm{Tr}\left( \left\vert D_{z}\Phi\right\vert
^{2}-\left\vert D_{\bar{z}}\Phi\right\vert ^{2}\right) , \label{Q1}%
\end{equation}
and%

\begin{equation}
E=\mathrm{Tr}\left( \left| D_{z}\Phi\right| ^{2}+\left| D_{\bar{z}}%
\Phi\right| ^{2}\right) \geq2\pi\left| Q\right| . \label{E1}%
\end{equation}
The configuration which saturates the energy bound satisfies the BPS soliton
equation
\begin{equation}
D_{\bar{z}}\Phi=0, \label{bps1-1}%
\end{equation}
or BPS anti-soliton equation
\begin{equation}
D_{z}\Phi=0. \label{bps2-1}%
\end{equation}


\smallskip It is convenient \cite{Lee:2000ey} to introduce the 2-component
complex vector $W$ and the projection operator $P$ as
\begin{equation}
\Phi=W\frac{1}{\sqrt{W^{\dagger}W}},\ \ P=\Phi\Phi^{\dagger}.
\end{equation}
In terms of the projection operator, BPS soliton equations
(\ref{bps1-1}) and
(\ref{bps2-1}) are
\begin{equation}
\left( 1-P\right) zP=0,\label{bps1-2}%
\end{equation}
and%

\begin{equation}
\left( 1-P\right) \bar{z}P=0. \label{bps2-2}%
\end{equation}
Topological charge (\ref{Q1}) and static energy (\ref{E1}) can be expressed
as
\begin{equation}
Q=\frac{1}{2\pi}\text{Tr}\left\{ \partial_{\bar{z}}\Phi^{\dagger}\left(
1-P\right) \partial_{z}\Phi-\partial_{z}\Phi^{\dagger}\left( 1-P\right)
\partial_{\bar{z}}\Phi\right\} , \label{Q2}%
\end{equation}
and%

\begin{equation}
E=\text{Tr}\left\{ \partial_{\bar{z}}\Phi^{\dagger}\left( 1-P\right)
\partial_{z}\Phi+\partial_{z}\Phi^{\dagger}\left( 1-P\right) \partial
_{\bar{z}}\Phi\right\} . \label{E2}%
\end{equation}


The examples of BPS soliton are $W=(z^{n},1)^{t}$ , with topological charge
$Q=n$ and energy $E=2\pi n$, and those of BPS anti-soliton are $W=(\bar{z}%
^{n},1)^{t}$ with topological charge $Q=-n$ and energy $E=2\pi n$
\cite{Lee:2000ey}. These configurations are solitons also in the
commutative theory.

\section[ New Solitons]{New Solitons}

\smallskip We have found that the following is the BPS soliton solution
of the
non-commutative $CP^{1}$ model. The configuration of the soliton with the
topological charge $Q=-n$ is%

\begin{equation}
\Phi=\left(
\begin{array}
[c]{c}%
\bar{z}^{n}\dfrac{1}{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}\\
0
\end{array}
\right) , \label{phi1}%
\end{equation}
which can also be expressed in terms of projection operator $P$ as%

\begin{equation}
P=\left(
\begin{array}
[c]{cc}%
1-\sum_{m=0}^{n-1}\left\vert m\right\rangle \left\langle m\right\vert & 0\\
0 & 0
\end{array}
\right) . \label{P1}%
\end{equation}
For $Q=n,$ on the other hand, the soliton can be written as%

\begin{equation}
\Phi=\left(
\begin{array}
[c]{c}%
\dfrac{1}{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}z^{n}\\
\sum_{m=0}^{n-1}\left\vert m\right\rangle \left\langle m\right\vert
\end{array}
\right) , \label{phi2}%
\end{equation}
and the corresponding projection operator expression is%

\begin{equation}
P=\left(
\begin{array}
[c]{cc}%
1 & 0\\
0 & \sum_{m=0}^{n-1}\left\vert m\right\rangle \left\langle m\right\vert
\end{array}
\right) . \label{P2}%
\end{equation}
We can straightforwardly confirm that (\ref{P1}) and (\ref{P2}) satisfy the
BPS equations (\ref{bps2-2}) and (\ref{bps1-2}) respectively. The energy of
these solitons are of course $2\pi n$, as is seen below.

We calculate the topological charge and static energy of (\ref{phi1}) and
(\ref{phi2}). For BPS anti-solitons (\ref{phi1}), we use the following
relations%
\begin{align}
& \partial_{\bar{z}}\Phi^{\dagger}\left( 1-P\right) \partial_{z}%
\Phi\label{as1}\\
& =\theta^{-2}\left[ \frac{1}{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}%
z^{n}\ ,\ z\right] \left( \sum_{m=0}^{n-1}\left\vert m\right\rangle
\left\langle m\right\vert \right) \left[ \bar{z}\ ,\ \bar{z}^{n}\frac
{1}{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}\right] \nonumber\\
& =0,\nonumber
\end{align}
%

\begin{align}
& \partial_{z}\Phi^{\dagger}\left( 1-P\right) \partial_{\bar{z}}%
\Phi\label{as2}\\
& =\theta^{-2}\left[ \frac{1}{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}%
z^{n}\ ,\ \bar{z}\right] \left( \sum_{m=0}^{n-1}\left\vert m\right\rangle
\left\langle m\right\vert \right) \left[ z\ ,\ \bar{z}^{n}\frac{1}%
{\sqrt{\prod_{l=1}^{n}(\bar{z}z+l\theta)}}\right] \nonumber\\
& =\theta^{-1}n\left\vert 0\right\rangle \left\langle 0\right\vert
.\nonumber
\end{align}


Substituting (\ref{as1}) and (\ref{as2}) into (\ref{Q2}) and (\ref{E2}), the
topological charge and static energy are%

\begin{equation}
Q=\frac{1}{2\pi}2\pi\theta\sum_{k=0}^{\infty}\left\langle k\right| \left(
-\theta^{-1}n\left| 0\right\rangle \left\langle 0\right| \right) \left|
k\right\rangle =-n, \label{Q3}%
\end{equation}
and%

\begin{equation}
E=2\pi\theta\sum_{k=0}^{\infty}\left\langle k\right| \left( \theta
^{-1}n\left| 0\right\rangle \left\langle 0\right| \right) \left|
k\right\rangle =2\pi n. \label{E3}%
\end{equation}
Similarly, for BPS solitons (\ref{phi2}), we use the relations%

\begin{align}
\partial_{\bar{z}}\Phi^{\dagger}\left( 1-P\right) \partial_{z}\Phi &
=\theta^{-2}\left[ \sum_{l=0}^{n-1}\left\vert l\right\rangle \left\langle
l\right\vert \ ,\ z\right] \left( 1-\sum_{m=0}^{n-1}\left\vert
m\right\rangle \left\langle m\right\vert \right) \left[ \bar{z}%
\ ,\ \sum_{l=0}^{n-1}\left\vert l\right\rangle \left\langle l\right\vert
\right] \label{S1}\\
& =\theta^{-1}n\left\vert n-1\right\rangle \left\langle n-1\right\vert
,\nonumber
\end{align}
%

\begin{align}
\partial_{z}\Phi^{\dagger}\left( 1-P\right) \partial_{\bar{z}}\Phi &
=\theta^{-2}\left[ \sum_{l=0}^{n-1}\left\vert l\right\rangle \left\langle
l\right\vert \ ,\ \bar{z}\right] \left( 1-\sum_{m=0}^{n-1}\left\vert
m\right\rangle \left\langle m\right\vert \right) \left[ z\ ,\ \sum
_{l=0}^{n-1}\left\vert l\right\rangle \left\langle l\right\vert \right]
\label{S2}\\
& =0.\nonumber
\end{align}
Then, the topological charge and static energy are $Q=n$ and $E=2\pi n$
respectively.

\section[ rel. to Comm. Space]{Relation to the Commutative Case}

We note that these solutions do not have the commutative counterparts. In
order to see this, we consider
\begin{equation}
W=\left(
\begin{array}
[c]{c}%
a^{-n}\bar{z}^{n}\prod_{l=1}^{n}\left( \bar{z}z+l\theta\right) ^{-1}\\
1
\end{array}
\right) , \label{w1}%
\end{equation}
for $Q<0$, where $a$ is a real parameter. Taking the limit
$a\rightarrow0$ in
the non-commutative case, $\Phi$ reduces to the non-singular configuration
(\ref{phi1}). On the other hand, in the commutative case, the same limit
leads
to
\begin{equation}
\Phi=\left(
\begin{array}
[c]{c}%
e^{-in\varphi}\\
0
\end{array}
\right) ,
\end{equation}
here$\ z\equiv\left\vert z\right\vert e^{i\varphi}\ $. This configuration is
ill-defined at the origin.

The case of $Q>0$ goes along the similar line. We consider
\begin{equation}
W=\left(
\begin{array}
[c]{c}%
a^{-n}\left( \prod_{l=1}^{n}\left( \bar{z}z+l\theta\right) ^{-1}\right)
z^{n}\\
1
\end{array}
\right) , \label{w2}%
\end{equation}
and taking the limit $a\rightarrow0$ in non-commutative case, $\Phi$ reduces
to (\ref{phi2}) which is non-singular. On the other hand, in the commutative
case, the same limit leads to
\begin{equation}
\Phi=\left(
\begin{array}
[c]{c}%
e^{in\varphi}\\
0
\end{array}
\right) ,
\end{equation}
which is also ill-defined at the origin. This shows that in the commutative
theory the configurations corresponding to $a\rightarrow0$ limit of
(\ref{w1}), (\ref{w2}) are absent.

Thus, we have seen that the singular configurations in the commutative
theory
become non-singular in the non-commutative case which can be attributed to
non-commutativity of the coordinates. It should be noted here that the
solutions investigated in \cite{Furuta:2002nv} using the coherent states are
different from ours. Although their special case of $\left(
\begin{array}
[c]{cc}%
\left\vert 0\right\rangle \left\langle 0\right\vert & 0\\
0 & 1
\end{array}
\right) $ does coincide with our particular case of BPS solitons in
(\ref{P2}).

To summarize, we have considered solitons in the non-commutative $CP^{1}$
model and have found new solitons that do not have their counterparts in the
commutative theory. Further properties of these solutions will be examined
using more general
approaches\cite{Hamanaka:2000aq}\cite{Hamanaka:2001dr} in a
future publication.

\providecommand{\href}[2]{#2}\begingroup \raggedright

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