ref{eq:lag_lsm}) satisfy 
\beq
\eta^2 > 0,\quad 3 \lambda_1 + \lambda_2 > 0.
\eeq
 Up to the flavor rotation, one can choose the ground state value as
\beq
\left<\Sigma\right> = v {\bf 1}_3,\quad v = \sqrt{\frac{\eta^2}{2(3\lambda_1 + \lambda_2)}}.
\eeq
The vacuum expectation value $v$ should be identified with 
$v \sim f_\pi \sim f_{\eta'}$.
In the ground state, the chiral symmetry $G_{\rm F}$ breaks down to
its diagonal subgroup $H_{\rm F} = SU(3)_{\rm V}/\mathbb Z_{3{\rm V}}$ given in Eq.~(\ref{eq:HF}). 
Then, the order parameter manifold is 
$G_{\rm F}/H_{\rm F} \simeq  U(3)_{\rm L-R+A}$ given in Eq.~(\ref{eq:GFHF}). 

The mass spectra are as follows: there are $8+1$ NG bosons associated with the spontaneously 
broken $U(3)_{\rm L-R+A}$ and the same number of massive bosons, whose masses are
\beq
m_1^2 = 2\eta^2,\quad
m_8^2 = 4\lambda_2v^2.
\eeq


Note that the linear sigma model (\ref{eq:lag_lsm}) can be regarded as that for the chiral symmetry breaking 
in the low density hadronic phase.
There are a number of works studying various topological solitons in the hadronic phase.
For example, $\eta'$ strings, which are 
the $U(1)_{\rm A}$  global vortices, were studied
in Refs.~\cite{Zhang:1997is,Balachandran:2001qn}. 
Non-topological vortices called pion strings with a 
trivial topology $\pi_1 [SU(2)]=0$ 
were studied in Refs.~\cite{Brandenberger:1998ew,Zhang:1997is}. 

Non-Abelian global vortices in the hadronic phase were 
first found by Balachandran and Digal in Ref.~\cite{Balachandran:2002je}.
Various aspects of the non-Abelian global vortices were studied 
in the subsequent papers \cite{Nitta:2007dp,Nakano:2007dq,Eto:2009wu}.
Since the linear sigma model for the chiral symmetry breaking in the CFL phase 
shares many similar properties with that for the hadronic phase,
there should exist the same kind of non-Abelian global vortices
in dense QCD.


\subsection{Abelian and non-Abelian axial vortices}
\label{sec:global_vortex}

\subsubsection{Abelian axial vortices}\label{sec:gv_A}

Since the order parameter manifold 
$G_{\rm F}/H_{\rm F} \simeq U(3)_{\rm L-R+A}$ is not simply connected, the first homotopy group is nontrivial: 
\beq 
 \pi_1 [U(1)_{\rm A}] \simeq \mathbb{Z}. \label{eq:A-gl-vor-homotopy}
\eeq
Therefore, there exist topologically stable vortices.
In order to generate a nontrivial loop in 
the order parameter manifold, 
one may use only a $T_0$ generator.
Such a loop corresponds to the $\eta'$ string \cite{Zhang:1997is,Balachandran:2001qn}
for which the order parameter $\Sigma$ behaves as
\beq
\Sigma(r,\theta) = v f(r) e^{i\theta}{\bf 1}_3.
\eeq
The equation of motion for the amplitude function $f(r)$ is of the form
\beq
u^2\left(f'' + \frac{f'}{r} - \frac{f}{r^2}\right) - \frac{m_1^2}{2}f(f^2-1)=0.
\eeq
This is exactly the same equation as the equation of motion for the familiar $U(1)$ global vortices.
Solutions can be numerically obtained with the suitable boundary condition
$f(0)=0$ and $f(\infty) = 1$; see {\rm e.g.} Ref.~\cite{Vilenkin:1994}.
The tension of the Abelian global string is \cite{Nitta:2007dp}
\beq
T_{U(1)_{\rm A}} = 3 \times 2\pi v^2u^2 \log\frac{L}{\xi} + \text{const.}\,.
\eeq
with the size of the system $L$ and the size of the axial vortex 
$\xi = m_1^{-1}$.


\subsubsection{Non-Abelian axial vortices}\label{sec:gv_NA}
One can construct a smaller loop inside the order parameter manifold by combining the $U(1)_{\rm A}$ generator 
$T_0 \sim {\bf 1}_3$ and non-Abelian generators 
$T_a$ ($a=1,2,\cdots,8$) of $SU(3)$ \cite{Balachandran:2002je}. 
They correspond to non-Abelian axial vortices 
characterized by the homotopy group
\beq 
 \pi_1 [U(3)_{\rm L-R+A}] \simeq \mathbb{Z}. \label{eq:NA-gl-vor-homotopy}
\eeq
The three typical configurations are given by 
\beq
&& \Sigma = v\, {\rm diag}\left(e^{i\theta}f(r),g(r),g(r)\right)
\xrightarrow[]{r\to\infty} 
v\,{\rm diag}\left(e^{i\theta},1,1\right)
=v\,e^{i\frac{\theta}{3}} {\rm diag}\left(e^{i\frac{2\theta}{3}},e^{-i\frac{\theta}{3}},e^{-i\frac{\theta}{3}}\right), 
 \non
&& \Sigma = v\, {\rm diag}\left(g(r),e^{i\theta}f(r),g(r)\right)
\xrightarrow[]{r\to\infty} 
v\,{\rm diag}\left(1,e^{i\theta},1\right)
=v\,e^{i\frac{\theta}{3}} {\rm diag}\left(e^{-i\frac{\theta}{3}},e^{i\frac{2\theta}{3}},e^{-i\frac{\theta}{3}}\right),
\non
&& \Sigma = v\, {\rm diag}\left(g(r),g(r),e^{i\theta}f(r)\right)
\xrightarrow[]{r\to\infty} 
v\,{\rm diag}\left(1,1,e^{i\theta}\right)
=v\,e^{i\frac{\theta}{3}} {\rm diag}\left(e^{-i\frac{\theta}{3}},e^{-i\frac{\theta}{3}},e^{i\frac{2\theta}{3}}\right).\non
\label{eq:ansz_ngv}
\eeq
with the profile functions $f(r)$ and $g(r)$ satisfying 
the boundary conditions
\beq
 f(r\to \infty) = g(r\to \infty) = 1,
\quad 
f(0) = g'(0)=0.
\eeq
The boundary condition at spatial infinities 
in Eqs.~(\ref{eq:ansz_ngv}) 
clearly shows that the corresponding loops 
wind 1/3 of the $U(1)_{\rm A}$ phase, 
and are generated by 
non-Abelian generators of $SU(3)_{\rm L-R}$ 
at the same time. 
They are called fractional vortices 
because of the fractional winding of the $U(1)_{\rm A}$ phase
and non-Abelian vortices  
because of the contribution of the non-Abelian generators. 
Because of the fractionality of the $U(1)_{\rm A}$ winding, 
the tension of a single non-Abelian axial vortex is one-third of that 
for an Abelian axial vortex \cite{Nitta:2007dp}:
\beq
T_{U(3)_{\rm L-R+A}} = 2\pi v^2 u^2 \log\frac{L}{\xi} + \text{const.}\,,
\eeq
with $\xi\sim {\rm max}(m_8^{-1},m_1^{-1})$.

The equation of motion for the amplitudes $f(r)$ and $g(r)$ are
\beq
&&u^2\left(f'' + \frac{f'}{r} - \frac{f}{r^2}\right) - \frac{