\#1[ ==]{}\#1\#2 \#3 \#4 \#5\]\#6[\#2 \#3 \#4 \#5 .\
]{}\#1\#1[=\#1 [=12 \#\#1[‘\#\#1=]{}‘ =10 to :.\
]{}]{}\#1[ = by- = by- =0 =0 == = = - = 2 &gt;0 0 0$ because the rest are related
to the $n0$ ones by a field
redefinition. It can also be arranged, without loss of generality, for the
coupling constant
$$ to lie in the Cartan subalgebra of the group $G$.  With
this choice for $$, we shall show that all vacua lie on a
maximal torus of the group $G$.   We shall
first solve the Bogomol'nyi equations of $SO(2)$-models and show that
the $SO(2)$ solitons generalise the sine-Gordon solitons; the latter
are recovered in
the limit $=0$. We shall then give the soliton solutions of the
$SU(2)$ model for $n=1$ and $n=2$ without
posing any restriction on the coupling constant $$.  In the
$n=1$ case, we shall find two classes of
solitons.  The first one is a set of static solitons that lie on a
maximal torus of $SU(2)$
and are embeddings of the $SO(2)$ solitons.  The second one is a set
of time-dependent solutions that, apart
from their asymptotic values, lie in the complement of the maximal
torus of $SU(2)$ and include the solution
given in [\gptb]. In the
$n=2$ case, we shall show that all solitons which lie on the maximal  
torus are obtained from embedding
of $SO(2)$ solitons into the $SU(2)$ model. 
In addition, we shall examine the qualitative properties of the Bogomol'nyi
equations for $SU(2)$  $n&gt;2$ models and support our conclusions with a
numerical calculation for the $n=3, 4, 5$ cases (Fig [1,2,3]).
Finally, we shall show that the $G$-model for
$G$ any semisimple Lie group has static and time-dependent solutions.
The former lie on the maximal torus
of $G$ while the latter lie on the complement of the maximal torus.
Some of the $G$-model solitons 
are obtained from embedding $SO(2)$ and $SU(2)$ solitons.

The organisation of the paper is as follows:  In section two, we shall
introduce the Bogomol'nyi equations of
$G$-models and discuss their general properties. In section
three,  we shall solve the Bogomol'nyi equations for $G=SO(2)$ and
compute the charge, mass and moduli space
metric of the solitons. In section four, we shall examine the general
properties of the $SU(2)$ model
and give all static solitons that lie on the maximal torus of $SU(2)$.
In section five, we shall investigate
the time-dependent solitons  of the
$SU(2)$ $n=1$ model and give their charge, mass and moduli space
metric.  In section six, we shall
examine the solitons of the $SU(2)$ $n=2$ model and find that all of
them are static and lie on the maximal
torus. In section seven, we shall study the qualitative properties of
the Bogomol'nyi equations for the
$SU(2)$ $n&gt;2$ models. The solitons for
models with any semisimple group $G$ as target space will be
investigated in section eight, and in section nine we
shall give our conclusions and comment on the quantum mechanical
properties of the solitons of $G$-models.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter { The Bogomol'nyi equations}

Let $[N]{}$ be a Riemannian manifold with metric $g$, a locally
defined two form $b$ and a Killing vector
$X$. The bosonic part of the action of (1,1)-supersymmetric massive
sigma models with torsion $H$ and scalar
potential
$V$ [\gptb]  is
$$
I= \int d^2x \big[ (g+b)_{ij} \partial_+ \phi^i \partial_-  
\phi^j-V(\phi)\big]\ ,
\eqn\onebb
$$
where $$ is a map from the two-dimensional Minkowski spacetime,
with light-cone co-ordinates
$(x\^+, x\^-)$, into $[N]{}$ and the two-form $b$ is the `gauge
potential' of the
torsion three-form $H$, $H\_[ijk]{}=[32]{}\_[\[i]{} b\_[jk\]]{}$.
Furthermore, the scalar potential is
$$
V={m^2\over4} g_{ij} \big( X^i X^j+  u^i u^j\big)\ ,
\eqn\potone
$$
where the one-form $u$ is orthogonal to the Killing vector $X$ ($X\^i u\_i=0$) and
$$
X^i H_{ijk}=\partial_{[j} u_{k]}\ .
\eqn\pottwo
$$
It is clear that the supersymmetric vacua of the theory are the points
of the target space $[N]{}$ where
both
$X$ and $u$ vanish. The action of the (1,1)-supersymmetric massive
sigma model is invariant under the
transformations
$$
\delta\phi^i=\eta X^i(\phi)
\eqn\ptthree
$$
with associated charge 
$$
Q=\int \, dx\, \big(X_i\,\partial_t\phi^i+u_i\, \partial_x\phi^i\big)\ ,
\eqn\potfour
$$
 where $$ is an infinitesimal parameter and  $(t,x)$ are the
co-ordinates of two-dimensional Minkowski spacetime. 
The  charge $Q$ is the charge that appears as central charge in the
(1,1)-supersymmetry
algebra [\gptb].  The energy of the model is
$$
E={1\over2} \int\, dx\, \big(g_{ij}\, \partial_t\phi^i\,  
\partial_t\phi^j+g_{ij}\, \partial_x\phi^i\,
\partial_x\phi^j+V(\phi)\big)\ ,
\eqn\fivebb
$$
and the  bound of $E$ in terms of $Q$ is
$$
E\geq {m\over2} |Q|\ .
\eqn\sixbb
$$
We can split the charge $Q$ into a Noether charge $Q\_N$ and a  
topological charge $Q\_T$, $Q=Q\_N+Q\_T$. Although
there is no natural way to split $Q$, we can use the Hodge decomposion
of $u$ as a sum of an exact form
$$, a co-exact form
$$ and a harmonic form
$$, ($u=++$), with respect to the metric $g$ to define
$$
Q_T=\int(\alpha+\gamma)_i\partial_x\phi^i\ .
\eqn\potsix
$$
and $Q\_NQ-Q\_T$;  $Q\_N$ and $Q\_T$ are separately
conserved. Both the fundamental states and the solitons of the theory
are charged with respect to the Noether
charge $Q\_N$ but only the latter are charged with respect to the    
topological charge $Q\_T$ as well.


In the special case of $G$-models with scalar potential \onea, we
choose the couplings as follows:
$g$ is the bi-invariant metric 
$$
g_{ij}=\delta_{AB}L^A{}_i L^B{}_j=\delta_{AB}R^A{}_i R^B{}_j
\eqn\metrone
$$
on $G$,
$H$ is the bi-invariant closed three form 
$$
H_{ijk}=-{1\over2}f_{ABC}L^A_iL^B_jL^C_k=-{1\over2} f_{ABC} R^A_iR^B_jR^C_k
\eqn\twobb
$$
on $G$, $X$ is the vector field
$$
X^i=\kappa^A (L-R)^i_A\ ,
\eqn\potseven
$$
and the one-form $u$ is
$$
u_i=\kappa_A (L+R)_i^A+\partial_ia\ ,
\eqn\poteight
$$
where $L$ is the left invariant frame of $G$, $R$ is the right
invariant frame of $G$, $f$ are the
structure constants of $[L]{}(G)$, $A,B,C=1,...,[dim]{}[L]{}(G)$ and 
$i,j,k=1,...,[dim]{} G$, $$ is a vector in the Lie algebra
of $G$ as in \onea\ and $a$ is given
in \oneaa.

The supersymmetric classical vacua of the action \onebb\ are the
solutions of the algebraic equation
$$
2\kappa_A+{\rm tr}\big( k^n t_A\big)=0
\eqn\fourb
$$
for $kG$,
where ${t\_A; A=1,...,[dim]{}[L]{}(G)}$ is a basis of the Lie
algebra of the group $G$.
The charge $Q$ \potfour\ of $G$-models is
$$
Q=\int dx \big(\kappa_A (L-R)^A_i\partial_t\phi^i+\kappa_A  
(L+R)^A_i\partial_x\phi^i +\partial_x a \big)\ .
\eqn\fourbb
$$
For $G$ abelian, $L=R$ and $\_A(L+R)\^A$ is a closed one-form, so
$Q\_N=0$ and
$$
Q_T=Q=\int \, dx\, \big(2\kappa_A L^A_i\partial_x\phi^i +\partial_x a \big)\ .
\eqn\potninea
$$  
However, for $G$ semisimple, the one-form $u-da$ is  co-exact and the
topological charge $Q\_T$ \potsix\  is
simply
$$
Q_T=\int dx \partial_x a\ .
\eqn\potnine
$$

To find the mass and charge of the bosonic fundamental  states of
$G$-models, we use the background field method (see for example
[\hps]) to linearise the theory around a
classical supersymmetric vacuum $k\_0$. The mass-square and charge    
matrices written in the left-invariant
frame basis are then
$$
{\cal M}_{AB}={m^2\over4}\big(\kappa^C \kappa^D f^E{}_{AC}\, f_{EBD}+
n^2 \sum_C{\rm tr}\,(t_{(A}\, t_{C)}
k_0^n)\,{\rm tr}(t_{(B}\, t_{C)} k_0^n)\big)\ ,
\eqn\potten
$$
and
$$
{\cal Q}_{AB}=-\kappa^C f_{ABC}\ ,
\eqn\poteleven
$$
respectively. The eigenvalues of  $[M]{}$ are the  square of the 
masses of the fundamental
states and the eigenvalues of $[Q]{}$ are their charges.  To do
this computation, we have used
\pottwo\ to show that
$du=0$ evaluated at any supersymmetric vacuum.  The fundamental states
parallel to the zero modes of the charge-matrix
\poteleven\ have zero Noether charge. The zero modes of $[Q]{}$
are orthogonal to the tangent vectors of
the coadjoint orbit of $G$ determined by $$ and therefore
the directions that carry the charge are those
along the coadjoint orbit; in fact $[Q]{}$ is the symplectic form
of the coadjoint orbit. The mass of the
fundamental states need not saturate the bound \sixbb\ since
 $$
{\cal M}\geq {m^2\over 4} {\cal Q} {\cal Q}^t
\eqn\potelevenb
$$
due to the presence of the second term in the mass-square-matrix
\potten. So the fundamental states obey the
bound as expected for supersymmetric theories but they are not  
necessarily BPS states.  Apart from the bosonic
fundamental states, the (1,1)-supersymmetric sigma models have also 
fermionic ones which can be investigated
in a similar way; for the purpose of this paper we shall restrict our 
attention  to the bosonic fundamental states. 


The classical configurations that saturate the bound \sixbb\ satisfy
the following Bogomol'nyi type equations:
$$
\eqalign{
\partial_t k(x,t)&=\mp {m\over2} (\kappa k-k \kappa)
\cr
\partial_x k(x,t)&=\mp {m\over2} \big(\kappa k+k \kappa +k \sum_A t_A
{\rm tr}(t_A k^n)\big)\ .}
\eqn\oneb
$$
The first of these equations can be easily solved by setting
$$
k(x,t)=\exp\big(\mp {m\over2} \kappa t\big) k(x) \exp\big(\pm
{m\over2} \kappa t\big)\ .
\eqn\twob
$$
Substituting this into the second equation of \oneb, it reduces to 
$$
{d \over dx}k(x)=\mp {m\over2} \big(\kappa k(x)+k(x) \kappa +k(x)
\sum_A t_A {\rm tr}(t_A
k^n(x))\big)\ .
\eqn\threeb 
$$
This is a non-linear but ordinary differential equation and the 
investigation of its solutions for various
choices of $G$ and $n$ will be our task in the rest of the paper. 

It is sufficient to investigate \threeb\ for
$n&gt;0$ because the transformation
$$
k\rightarrow k^{-1}, \qquad x\rightarrow -x 
\eqn\fiveba
$$
acting on a model with coupling constants $g, b,, m$  and
$a=[1n]{}[tr]{} k\^n$ transforms it to a
model with coupling constants $g, b,, m$ and
$a=-[1n]{}[tr]{} k\^[-n]{}$.  Therefore the solitons of models 
with $n&lt;0$  can be obtained from
the solitons of models with $n&gt;0$ by acting on the latter with \fiveba.  
In addition, the Lagrangian \onebb, the field equations and the
Bogomol'nyi equations of the
(1,1)-supersymmetric sigma model are invariant under the sigma model symmetry
$$
k(x,t)\rightarrow h k(x,t) h^{-1} \qquad \kappa\rightarrow h \kappa h^{-1}\ ,
\eqn\fiveb
$$
 where $hG$.  Although sigma
model symmetries are not associated with conserved charges, we can use
\fiveb\ to choose, without loss of
generality, the coupling constant $$ in the Cartan subalgebra of
$[L]{} (G)$. This choice
of $$ simplifies both the Bogomol'nyi equations \oneb\ and the 
equation for the vacua  \fourb\ of the
model. In particular, we shall show in chapter eight that it can be
arranged for the vacua to lie in the
maximal torus of the group $G$.

 
The solutions of the Bogomol'nyi equations may depend on several
moduli parameters.  In that case, as
for BPS monopoles [\manton], one can define a metric on the moduli
space of solutions as
follows:
$$
ds^2={1\over2}\int dx\, g_{ij} d\phi^i d\phi^j\ ,
\eqn\sevenb
$$
where the differential on $$ denotes variation with respect to the
moduli parameters. The soliton
solutions of the Bogomol'nyi equations \oneb\ preserve $1/2$ of the 
supersymmetry and therefore their
effective theory is an $N=1$ supersymmetric one-dimensional sigma
model with target space the moduli space and
metric the moduli metric \sevenb. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Chapter 2%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The SO(2) model}

The simplest of the $G$-models is the one with the group $SO(2)$ as
target space.
Since the target space is one dimensional, the antisymmetric tensor
coupling $b$ is identically zero.  A
parameterisation of the group manifold
$SO(2)$ is 
$$
k=\pmatrix{\cos\theta & \quad -\sin\theta\cr \sin\theta & \quad \cos\theta}
\eqn\onec
$$
and a basis in $[L]{}(SO(2))$ is
$$
t_1=\pmatrix{0&\quad -1\cr 1&\quad 0}\ ,
\eqn\onecnew
$$
where $01$, the model does not have any supersymmetric vacua and
therefore supersymmetry is spontaneously
broken. For $01$, the supersymmetric vacua of the
$SO(2)$ model are
$$
\theta[\tilde \ell, \pm]={1\over n}\big(\pm\theta_0+ {1\mp 1\over 2} 
\pi + 2\pi \tilde\ell\big) \ , 
\eqn\threec
$$
where $$ is an integer,  $00 ]{} $$ interpolate between the vacua $\theta[\ell, -]$ and $\theta[ \ell+1, +]$, for $0\leq\ell<n-1$, and the vacua $\theta[n-1, -]$ and $\theta[0, +]$ for $\ell=n-1$. We remark that the solutions  are continuous and differentiable at $x=0$. The solitons are those solutions in  and in  with the plus sign whereas the anti-solitons are those with the minus sign. We note that the $SO(2)$ solitons of  and  interpolate between all neighbouring pairs of vacua of the model. (ii) For $\kappa=0$, the equation  becomes that of the sine-Gordon theory. The solutions are $$\theta=\zeta\, {2\over n}\bigg({\rm arc}\tan\big[\exp\big(\pm m n 
(x-x_0)\big)\big]+\ell \pi\bigg)
\eqn\sixc$$ for $0\leq \ell< n$, $\ell\in \bZ$ and $\zeta=\pm$. (iii) For $\kappa=1$ and $n>1$ , the solutions that interpolate between the vacua $\theta[\ell]$ and $\theta[\ell+1]$, for $0\leq\ell<n-1$, and the vacua $\theta[n-1]$ and $\theta[0]$ for $\ell=n-1$, are $$\theta={2\over n}\bigg({\rm arc}\tan\big[\mp m n (x-x_0)\big]-
{\pi\over4}+\ell \pi\bigg)\ . 
\eqn\sixcc$$ These are $2n$ solutions, $n$ solitons and $n$ anti-solitons, interpolating between all neighbouring pairs of vacua as in the previous two cases.

For $SO(2)$ models, the charge $Q$  is equal to the topological charge $Q_T$ . The value of $Q$ for the solitons is $$Q=\mp {4\over n} \bigg[\sqrt{1-\kappa^2}+\kappa\, {\rm arc}
\sin(\kappa)+\xi\, {\kappa\over2}\pi\bigg]
\ ,
\eqn\sevenc$$ where $\xi$ is either minus or plus, ($\xi=\mp$); $Q$ for $\xi=-$ is the charge of  solutions and $Q$ for $\xi=+$ is the charge of  solutions. From this, it is straightforward to compute the mass of the solitons to find $$E={2 m\over n}\bigg[\sqrt{1-\kappa^2}+\kappa\, {\rm
arc}\sin(\kappa)+\xi\, {\kappa\over2}\pi\bigg]\ .
\eqn\eightc$$ The $SO(2)$ solitons  and  have one moduli parameter $x_0$. Using , the metric on the moduli space is $$ds^2={ m\over n}\bigg[\sqrt{1-\kappa^2}+\kappa\, {\rm arc}
\sin(\kappa)+\xi\, {\kappa\over2}\pi\bigg]
dx_0^2={E\over 2} dx_0^2\ .
\eqn\ninec$$ The charge, the mass and the moduli metric of the solitons   are derived by setting $\kappa=0$ in the above corresponding expressions.

Finally, the charge, the mass and the moduli metric of the solutions   of models with $\kappa=1$ are $$Q=\mp {4\over n} \pi\ ,
\eqn\tenc$$ $$E={2 m\over n} \pi\ ,
\eqn\tencc$$ and $$ds^2={m\over n} \pi dx_0^2={E\over2} dx_0^2\ ,
\eqn\tenccc$$ respectively.

The $SU(2)$ model
=================

A parameterisation of $SU(2)$ in terms of $2\times2$ matrices is $$k=\pmatrix{M+iN &\quad -U+iW \cr U+iW &\quad M-iN}; \quad M^2+N^2+U^2+W^2=1\ ,
\eqn\oned$$ and a choice of basis in the Lie algebra ${\cal L}\big(SU(2)\big)$ of $SU(2)$ is $$t_1=i\pmatrix{1&\quad 0\cr 0&\quad -1}\ , \quad t_2=i
\pmatrix{0&\quad 1\cr 1&\quad 0}\ ,\quad
t_3=\pmatrix{0&\quad -1\cr 1&\quad 0}\ .
\eqn\twod$$ Using the observation in section 2 that the Bogomol’nyi equations are invariant under the adjoint action of $G$ on both the fields and the coupling constant $\kappa$, we can choose, without loss of generality, $\kappa$ in the Cartan subalgebra of ${\cal L}\big(SU(2)\big)$, [i.e.]{} $$\kappa=\kappa t_1\ .
\eqn\threed$$ We have denoted both the vector $\kappa$ and its component along $t_1$ with the same symbol. The distinction between the two will be clear from the context.

The vacua of the theory   in the above parameterisation are the solutions of the equation $$k^n=\pmatrix{\mp \sqrt{1-\kappa^2}+i\kappa &\quad 0\cr 0&\quad \mp 
\sqrt{1-\kappa^2}-i\kappa}\ ,
\eqn\fourdn$$ in terms of $k$, $n\in \bN$. It is clear that for $|\kappa|>1$, there are no supersymmetric vacua and so supersymmetry is spontaneously broken. For $|\kappa|<1$ there are $2n$ supersymmetric vacua, and for $|\kappa|=1$ there are $n$ supersymmetric vacua (without counting multiplicities in the latter case). We shall restrict our attention to $\kappa\geq0$ because it is straightforward to generalise the results below to $SU(2)$ models with $\kappa<0$. We remark that linearising the theory around the vacua , the mass-square-matrix and charge-matrix of the fundamental states of the $SU(2)$ model are $${\cal M}=m^2 \pmatrix{&1-\kappa^2\quad  &0\quad  &0\cr
                &0\quad &1\quad &0\cr
                &0\quad &0\quad &1}\ ,
\eqn\massmatone$$ and $${\cal Q}=\pmatrix{&0\quad &0\quad &0\cr
                &0\quad &0\quad &-2\kappa\cr
                &0\quad &2 \kappa\quad &0}\ ,
\eqn\chargematone$$ respectively. Clearly for $|\kappa|<1$, ${\cal M}>{m^2\over 4} {\cal Q} {\cal Q}^t$ and the fundamental states are not BPS states. However, for $|\kappa|=1$, ${\cal M}={m^2\over 4} {\cal Q} {\cal Q}^t$ and all the fundamental states become BPS with one of them massless.

The equations  in this parameterisation for the $SU(2)$ model are $$\eqalign{
{d\over dx}M\pm m [-\kappa N+(1-M^2) \tilde M_n]&=0
\cr
{d\over dx}N\pm m[\kappa M-NM\tilde M_n]&=0
\cr
{d\over dx}U\mp m U M \tilde M_n&=0
\cr
{d\over dx}W\mp m W M \tilde M_n&=0\ ,}
\eqn\fivedn$$ where $$\tilde M_n=\sum^{[{n-1\over2}]}_{i=0}\pmatrix{n\cr 2i+1} 
(-1)^{i} M^{n-2i-1} (1-M^2)^{i}\ .
\eqn\sixdn$$ There are three independent equations in  since the fourth one is implied from the restriction that $M, N, U, W$ lie on a 3-sphere. Moreover from the last two equations of , it is easy to see that $$U=\lambda W\ ,
\eqn\sevendn$$ where $\lambda$ is a real constant. So we only need to determine $M,N$ from the first two equations in ; these equations will be studied in the next three chapters for various values of $n$.

It is convenient for reasons that will become apparent later to parameterise $SU(2)$ in angular coordinates as follows: $$\eqalign{
M&=\cos\theta
\cr
N&=\sin\theta \cos\phi
\cr
U&=\sin\theta  \sin\phi \cos\psi
\cr 
W&=\sin\theta \sin\phi \sin\psi\ ,}
\eqn\nineda$$ where $0\leq \theta, \phi<\pi$ and $0\leq \psi<2\pi$. In this parameterisation, the equation  for the vacua can be written as $$\eqalign{
\sin(n\theta)&=\pm \kappa
\cr
\cos\phi&=\pm 1\ . }
\eqn\nfived$$ Note that the vacua of the theory lie in $\phi=0$ and $\phi=\pi$ semi-circle subspaces of $SU(2)$, which are joined at $(\theta, \phi)=(0,0)$ and $(\theta, 
\phi)=(0,\pi)$, and at $(\theta,
\phi)=(\pi,0)$ and $(\theta, \phi)=(\pi,\pi)$ to a circle in $SU(2)$; in fact this circle is a maximal torus of $SU(2)$. Solving  for $0\leq \kappa <1$, we find that the $2n$ vacua of the model are $$\big(\theta, \phi\big)=\cases{\bigg({1\over
n}(\pm\theta_0+{1\mp1\over2}\pi+2\pi \tilde \ell), 0\bigg)\ ,
\qquad\qquad 0\leq\tilde
\ell\leq \big[{n-1\over2}\big]
\cr
\bigg({1\over n}\big(\pm\theta_0+{1\mp1\over2}\pi+2\pi (\tilde 
\ell+{1\over2})\big), \pi\bigg)\ ,\qquad 0\leq\tilde
\ell<\big[{n\over2}\big]\ ,}
\eqn\vacthree$$ where $0\leq \theta_0<{\pi\over2}$ and $\sin \theta_0=\kappa$. For $\kappa=0$, $\theta_0$ vanishes and the expression for the vacua becomes $$\eqalign{
(\theta, \phi)&=\big({\ell \over n}\pi, 0\big)
\cr
(\theta, \phi)&=\big({\ell+1 \over n}\pi, \pi\big)\ , }
\eqn\qnew$$ where $0\leq \ell<n$, $\ell \in \bN$. For $\kappa=1$, the $n$ vacua of the theory (without counting multiplicities) can be found by setting $\theta_0=\pi/2$ in .

The equations  written in the angular parameterisation are $$\eqalign{
{d\over dx}\theta&=\pm m[-\kappa \cos \phi+\sin(n\theta)]
\cr
{d\over dx}\phi&=\pm m\kappa \cot\theta \sin\phi
\cr
{d\over dx}\psi&=0\ .}
\eqn\ninedaan$$ The third equation above implies that the angle $\psi$ is constant; $\psi$ is related to the ratio of $U, W$ as $\cot\psi=\lambda$. So it remains to solve the first two equations of  for $\theta$ and $\phi$ for the various values of $n$ and $\kappa$.

To find a class of solutions for the equations  above, we set $\phi$ to its vacuum values, [i.e.]{} $\phi=0,\pi$. This choice for $\phi$ solves the second of the equations , so it remains to solve the first for $\theta$. Since we have set $\phi=0,\pi$, the solutions lie on the maximal torus of $SU(2)$, and they are best described by extending the range of $\theta$ from $0\leq
\theta<\pi$ in  to $0\leq \theta<2\pi$. Then, the first equation in  is precisely the equation  that we have encountered in the $SO(2)$ model. Moreover, the vacua  of the $SU(2)$ model can be identified with the vacua  of the $SO(2)$ model as follows: $$\theta[\ell,\pm]=\cases{\bigg({1\over
n}(\pm\theta_0+{1\mp1\over2}\pi+2\pi  \ell),
0\bigg)\ , \qquad\qquad 0\leq \ell\leq \big[{n-1\over2}\big]
\cr
\bigg({1\over n}\big(\pm\theta_0+{1\mp1\over2}\pi+2\pi (\tilde
\ell+{1\over2})\big), \pi\bigg)\ ,\qquad \big[{n-1\over2}\big]<
\ell<n \ ,}
\eqn\vacthreenew$$ where $\tilde \ell=n-\ell-1$. Furthermore, as can be easily seen from , the solitons of the $SU(2)$ model that lie on the maximal torus are static. From this we conclude that the solitons of the $SU(2)$ model that lie on the maximal torus are embeddings of the solitons of the $SO(2)$ model found in the previous section with $$\theta\rightarrow \pmatrix{e^{i\theta}& \quad 0\cr 0& \quad e^{-i\theta}}\ ,
\eqn\fourteend$$ for the associated value of $n$. The class of solutions that we have described includes those for which $\kappa$ vanishes. This is because, if $\kappa=0$, the second equation in  implies that $\phi$ is constant and the asymptotic behaviour of the solitons requires that it should be set to its vacuum values.

Note from  and  that the definition of the topological charge of the $G$-models with $G$ semisimple is different from the definition of the topological charge of the $SO(2)$ models. The value of the topological charge $Q_T$ of any soliton of the $SU(2)$ model, or any $G$-model with $G$ semisimple, interpolating between the two vacua $(\theta,\phi)$ and $(\theta', \phi')$ is $Q_T=a(\theta',
\phi')-a(\theta, \phi)$ while for the $SO(2)$ solitons $Q_T=Q$. Therefore the value of the topological charge of the $SO(2)$ solitons and the value of the topological charge of their embeddings as $SU(2)$ solitons are different. Apart from the $SU(2)$ solitons that lie on the maximal torus and we have examined above, the $SU(2)$ model has other solitons that interpolate between different vacua but otherwise lie entirely outside the maximal torus of $SU(2)$. We shall investigate these solitons in the following three sections.

The $SU(2)$ n=1 model
=====================

As we have seen in the previous section, there are static solitons that interpolate between the two vacua of $SU(2)$ $n=1$ model with $0\leq \kappa<1$ and lie entirely on the maximal torus of $SU(2)$. Here, we shall find another class of solitons for this model that interpolate between the two vacua but otherwise lie in the [*complement*]{} of the maximal torus of $SU(2)$. The equations  for $M,N$ in the $n=1$ case are $$\eqalign{
{d\over dx}M\pm m [-\kappa N+(1-M^2)]&=0
\cr
{d\over dx}N\pm m[\kappa M-NM]&=0
\ .}
\eqn\fived$$ Following \[\], we can find a class of solitons by setting $N=\kappa$. These solutions are $$\eqalign{
M&=\sqrt{1-\kappa^2} \tanh\big(\mp m \sqrt{1-\kappa^2}(x-x_0)\big)
\cr
N&=\kappa
\cr
U&=\sqrt{1-\kappa^2} \cosh^{-1}\big( m \sqrt{1-\kappa^2}(x-x_0)\big) \cos\psi
\cr
W&=\sqrt{1-\kappa^2} \cosh^{-1}\big( m \sqrt{1-\kappa^2}(x-x_0)\big) \sin\psi
\ ,}
\eqn\eightd$$ where $x_0, \psi$ are the modular parameters of the solution, $\cot\psi=\lambda$. It is clear that these solitons lie in the complement of the maximal torus since $U$ and $W$ do not vanish for any value of $x$ unless $x$ goes to infinity. The topology of the moduli space is $S^1\times \bR$: The modular parameter $x_0$ is due to the translational invariance of the underlying theory while the modular angular parameter $\psi$ is due to the charge $Q$ given in . These are reminiscent to the modular parameters of BPS monopoles in four-dimensions (see for example \[\]).

However, there is a more general class of $SU(2)$ solutions for which $N\not=\kappa$. To find these new solitons, we use the angular parameterisation and the equations   for $n=1$ become $$\eqalign{
{d\over dx}\theta&=\pm m[-\kappa \cos \phi+\sin(\theta)]
\cr
{d\over dx}\phi&=\pm m\kappa \cot\theta \sin\phi\ .}
\eqn\ninedaa$$ As it can been seen from , the $SU(2)$ $n=1$ model has two vacua $(\theta_0, 0)$ and $(\pi-\theta_0, 0)$. We remark that a linearisation of  at these two vacua reveals that one of them is a source while the other is a sink. The differential equations  is a Hamiltonian flow for the function $$\alpha(\theta, \phi)=-{1\over \sin\phi \sin\theta}+{1\over \kappa} \cot\phi
\eqn\hamone$$ with symplectic form $$\Omega=\pm  {1\over (m \,\kappa )\, \sin^2\phi \sin\theta}\, 
d\theta\wedge d\phi \ .
\eqn\hamtwo$$ Since $\alpha$ is preserved by the flow, we can rewrite  as $$\sin\theta={\kappa\over \cos\phi-\alpha \kappa \sin\phi}\ ,
\eqn\moned$$ where $\alpha$ is a real constant. The solutions of  in the $x$ parameterisation are $$\eqalign{
\cot\phi\equiv z&={\kappa\over 2 \sqrt{1-\kappa^2}}\exp{\big[\pm m 
\sqrt{1-\kappa^2} (x-x_0)\big]}
\cr & \qquad\qquad
\bigg([\exp{\big[\mp m \sqrt{1-\kappa^2} (x-x_0)\big]}+{\alpha\over 
\sqrt{1-\kappa^2}}]^2+1-\alpha^2\bigg)
\cr
\cos\theta&=\cases{{\sqrt{(z-\alpha\kappa)^2-\kappa^2(1+z^2)}\over 
z-\alpha\kappa}\ , \qquad -\infty< x
<x_{min}
\cr 
-{\sqrt{(z-\alpha\kappa)^2-\kappa^2(1+z^2)}\over z-\alpha\kappa}\ , 
\qquad x_{min}< x <+\infty\ ,}}
\eqn\mtwod$$ where at $x=x_{min}$, $z(x)$ takes its absolute minimum value. To verify that  solves the equations , we differentiate $z$ with respect to $x$ and use the equation for $\phi$ to eliminate the derivative of $\phi$ from the expression. The $x$ dependence can also be eliminated by inverting $z$ to express the exponential of $x$ in terms of $z$. However this equation is quadratic and there are two possible solutions distinguished by a sign. To satisfy the equations , one has to choose for $-\infty< x <x_{min}$ the solution with the plus sign and for $x_{min}< x <+\infty$ the solution with the minus sign together with the corresponding expression for $\cos\theta$ in .

To find the solitons of the $SU(2)$ $n=1$ model, we substitute  in  and observe that they are [*time-dependent*]{}. For $\alpha=0$,  reduces to the solution  given above. Moreover, after the redefinition $x_0\rightarrow x_0\pm {1\over
m\sqrt{1-\kappa^2}}\rm{log}(|\alpha| {1-\sqrt{1-\kappa^2}
\over\sqrt{1-\kappa^2}})$,  reduces in the limit $\alpha\rightarrow \pm \infty$ to the embedded SO(2) solitons found in the previous section. The charge and the mass of the solutions  are $$Q=\mp 4 \sqrt{1-\kappa^2} e^\sigma \ ,
\eqn\mthreed$$ and $$E= 2 m \sqrt{1-\kappa^2} e^\sigma\ ,
\eqn\mfourd$$ respectively, where $$e^\sigma=1+{\alpha \kappa^2\over \sqrt{1-\kappa^2} \sqrt{1+\alpha^2 
\kappa^2}}\bigg[- {\pi\over2}+{\rm
arc}\tan \big({\alpha \kappa^2\over \sqrt{1-\kappa^2} \sqrt{1+\alpha^2
\kappa^2}}\big)\bigg]\ .
\eqn\msixd$$ The constant $\alpha$ is not a modular parameter because the mass and the charge depend upon it. For example, the mass of the $\alpha=0$ solutions  is $$E=2 m \sqrt{1-\kappa^2}\ .
\eqn\msixdd$$ Note that the mass  of the solitons as a function of $\alpha$ has critical points at $\alpha\rightarrow\pm\infty$. It turns out that $E(+\infty)$ is the absolute minimum and is the value of the mass of the embedded static solution   for $n=1$, while $E(-\infty)$ is the absolute maximum and is the value of the mass of the embedded static solution   for $n=1$. This is in agreement with the corresponding limits of the solution  mentioned above. The moduli space of the solutions  is again a cylinder with co-ordinates $(x_0, \psi)$ and with metric $$ds^2=e^\sigma {\sqrt{1-\kappa^2}\over m
(1+\alpha^2 \kappa^2)}\bigg[m^2 
(1-\kappa^2+\alpha^2\kappa^2)  dx_0^2+ d\psi^2\bigg]\ .
\eqn\mfived$$ We remark that there is a transformation of the moduli co-ordinates $(x_0, \psi)\rightarrow (y, \chi)$ such that the moduli space metric above can be written in the form $ds^2={E\over2}(dy^2+d\chi^2)$.

The SU(2) n=2 model
===================

To show that all solitons of the $SU(2)$ $n=2$ model lie on the maximal torus of $SU(2)$, we begin from the equations  for $M,N$ with $n=2$ $$\eqalign{
{d\over dx}M\pm m [-\kappa N+2(1-M^2)M]&=0
\cr
{d\over dx}N\pm m[\kappa M-2NM^2]&=0 \ .}
\eqn\fived$$ We then use the field redefinitions $$X={N\over M}, \qquad Y={N\over \sqrt {1-M^2}}
\eqn\tend$$ to simplify  to $$\eqalign{
{d\over dx}X\pm  m\big[ \kappa (1+X^2)-2X\big]&=0
\cr
{d\over dx}Y\pm  m \kappa X^{-1} Y [1-Y^2]&=0\ .}
\eqn\elevend$$ The solutions in terms of $X,Y$ for $0<\kappa<1$ are $$\eqalign{
X&={1\over \kappa}\bigg(1+\sqrt{1-\kappa^2} \tanh\big(\pm m 
\sqrt{1-\kappa^2}(x-x_0)\big) \bigg)
\cr
{Y^2\over 1-Y^2}&={\rho}^{-1} X^2 \exp\big(\mp 2m (x-x_0)\big)\cosh^2\big(
m\sqrt{1-\kappa^2} (x-x_0)\big)\ ,}
\eqn\twelved$$ and $$\eqalign{
X&={1\over \kappa}\bigg(1+\sqrt{1-\kappa^2} \coth \big(\pm m 
\sqrt{1-\kappa^2} (x-x_0)\big) \bigg)
\cr
{Y^2\over 1-Y^2}&={\rho}^{-1} X^2 \exp\big(\mp 2m (x-x_0)\big) \sinh^2\big(
m\sqrt{1-\kappa^2} (x-x_0)\big)\ ,}
\eqn\twelved$$ where $\rho, x_0$ are real constants ($\rho\geq 0$). It remains to examine whether or not these solutions interpolate between the different vacua of the theory. As we have already mentioned in section four, the $SU(2)$ $n=2$ model has four vacua. It turns out that the solutions  interpolate between two different vacua only for $\rho=0$, in which case $U=W=0$. This implies that all solitons of the $SU(2)$ $n=2$ model lie on the maximal torus. The same applies for all solitons of the $SU(2)$ $n=2$ model with $\kappa=1$.

To explain why there are no solitons of the $SU(2)$ $n=2$ model that lie in the complement of the maximal torus of $SU(2)$, let us study the vacuum structure of this model in more detail. A linearisation of the equations  around the vacua   reveals that $({\theta_0\over2}, 0)$ and $({-\theta_0+2\pi \over2}, \pi)$ are sources, and $({\theta_0+\pi
\over2},\pi)$ and $({-\theta_0+ \pi \over2}, 0)$ are saddles (for the Bogomol’nyi equations with the plus sign). Solving the equations  with $n=2$ for $\theta$ and $\phi$ by setting $\phi=0,\pi$, we find precisely the solitons of the $SU(2)$ $n=2$ model that lie on the maximal torus of $SU(2)$ and therefore are embeddings of the $SO(2)$ solitons. These solitons interpolate between the source ($\phi=0$) and the saddle ($\phi=0$), the saddle ($\phi=0$) and the source ($\phi=\pi$), the source ($\phi=\pi$) and the saddle ($\phi=\pi$), and the saddle ($\phi=\pi$) and the source ($\phi=0$). Any additional solutions should interpolate either between the two saddles or between the saddles and the sources. But the only directions from the two saddles which are not connected to another vacuum are those that point outwards. Therefore no soliton can exist that starts from one saddle to go to the other or from a saddle to go to a source. In fact, the outward directions from the saddles and some of the directions from the sources are connected to a sink at $$(\theta, \phi)=({\pi\over 2}, {\pi\over2})\ ,
\eqn\fournew$$ which is a fixed point of the flow described by the equations , $n=2$, but [*not*]{} a vacuum of the theory. We shall conjecture in the next section that a similar behaviour occurs in all $SU(2)$ models with $n$ even.

The SU(2) $n>2$ models
======================

The equations  or  for $SU(2)$ models with $n>2$ and $\kappa\not= 0$ are rather involved and we have not been able to find the solitons that lie in the complement of the maximal torus of $SU(2)$. Another way to proceed is to investigate the qualitative properties of the solutions of these equations by linearising them about the vacua of the theory. This will allow us to find the pairs of vacua that are connected by solitons. The results of this analysis for $n=3,4,5$ have being confirmed with a numerical computation (see Figs. \[1,2,3\]).

It is sufficient to examine the linearisation properties of the equations  in the interval $0\leq
\theta <{\pi\over2}$. This is because the equations   are invariant under the discrete symmetries $$(\theta, \phi, x)\rightarrow (\pi-\theta, \pi-\phi, x)
\eqn\nthreed$$ for $n$ even, and $$(\theta, \phi, x)\rightarrow (\pi-\theta, \phi, -x)
\eqn\nfourd$$ for $n$ odd, which can be used to extend the analysis to the whole range of $\theta$, ($\theta\in [0,\pi)$). Another property of the equations  for $n$ [*even*]{} is that, apart from the vacua of the theory, there is another fixed point at $(\theta, \phi)=({\pi\over2}, {\pi\over2})$.

Next we order the vacua $(\theta,\phi)$ in  with respect to the value of $\theta$, and denote these values by $\theta_i$, $i=1,\dots, 
2n$, [i.e.]{} $\theta_1<\dots
\theta_i<\theta_{i+1}\dots<\theta_{2n}$. For $\theta$ in $[0,\pi/2)$, a linearisation of  with the plus sign reveals that the vacua $\theta_{4q+1}$ are sources, the vacua $\theta_{4q+3}$ are sinks and the rest are saddle points. Observe that the sources $\theta_{4q+1}$ and the saddles $\theta_{4q+2}$ lie at the $\phi=0$ semicircle, while the sinks $\theta_{4q+3}$ and the saddles $\theta_{4q+4}$ lie at the $\phi=\pi$ semicircle. First, we shall consider the solutions interpolating amongst vacua $\theta_i$ in the interval $[0,\pi/2)$. It is expected that there should be a one parameter family of solitons that interpolate between the source $\theta_{4q+1}$ and the sink $\theta_{4q+3}$, and the same source and the sink $\theta_{4q-1}$. There also should be a unique soliton interpolating between the source $\theta_{4q+1}$ and the saddle $\theta_{4q}$, and the sink $\theta_{4q+3}$ and the saddle $\theta_{4q+2}$. In addition to these solutions, we have the solitons that lie on the maximal torus. $$\vbox{
    \beginlabels\refpos 71 481 {}
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            \put 344 354 {\downarrow}
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            \put 134 309 {\theta_2}
            \put 134 401 {\theta_5}
            \put 134 447 {\theta_6}
            \put 419 331 {\theta_3}
            \put 419 377 {\theta_4}
        \put 130 462 {\pi}
            \put 124 355 {\pi/2}
            \put 146 239 {0}
        \put 408 239 {\pi} 
            \put 271 235 {\pi/2} 
            \put  99 348 {\uparrow}
            \put 268 218 {\rightarrow}
            \put 100 364 {\theta}
            \put 284 216 {\phi}
            \endlabels
            \epsfxsize=.65\hsize
            \centerline{\epsfbox{Fig1.eps}}}$$ -

[**Fig. 1:**]{} The solutions of the ${\rm SU(2)}$ $n=3$ model

The solitons interpolating amongst the vacua in $[\pi/2, \pi)$ are similar to those for the vacua in $[0,
\pi/2)$ examined above. This follows immediately from the discrete symmetries  and  of the equations . These symmetries act on $(\theta, \phi)$ that lie on the disc constructed from the square $[0,\pi]\times[0,\pi]$ by identifying the points $(0,0)$ and $(0,\pi)$, and the points $(\pi,0)$ and $(\pi,\pi)$. For $n$ even, the discrete symmetry is just the antipodal map of the disc leaving the centre $(\pi/2,\pi/2)$ fixed, whereas for $n$ odd, it is a reflection at the hyperplane $\theta=\pi/2$ together with a reversal of the direction of the flow and an interchange sinks and sources. $$\vbox{
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            \put 331 649 {\swarrow}
            \put 328 610 {\uparrow}
            \put 331 567 {\searrow}
            \put 326 528 {\nearrow}
            \put 132 512 {\theta_1}
            \put 132 540 {\theta_2}
            \put 419 554 {\theta_3}
            \put 419 581 {\theta_4}
            \put 132 595 {\theta_5}
            \put 132 623 {\theta_6}
            \put 419 637 {\theta_7}
            \put 419 664 {\theta_8}
            \put 132 678 {\theta_9}
            \put 132 705 {\theta_{10}}
        \put 128 716 {\pi}
            \put 122 609 {\pi/2}
        \put 146 493 {0}
        \put 408 493 {\pi} 
            \put 271 489 {\pi/2} 
            \put  99 602 {\uparrow}
            \put 268 472 {\rightarrow}
            \put 100 618 {\theta}
            \put 284 470 {\phi}
            \endlabels
            \epsfxsize=.65\hsize
            \centerline{\epsfbox{Fig2.eps}}}$$ -

[**Fig. 2:**]{} The solutions of the ${\rm SU(2)}$ $n=5$ model

Let us consider the solutions amongst the vacua that lie near the boundary of $[0,\pi]$. The lowest vacuum $\theta_1$ (which is a source) connects only to the sink $\theta_3$ through solitons that lie in the complement of the maximal torus of $SU(2)$. Similarly the highest vacuum $\theta_{2n}$ (which is a sink) connects only to the source $\theta_{2n-2}$. It remains to examine the solutions interpolating amongst vacua that lie ‘near’ $\theta=\pi/2$. For $n=4\ell+1$, $\ell\in \bN$, the highest vacuum in $[0,\pi/2)$ is the source $\theta_n$ and this connects to the sink $\theta_{n-2}$ and to the saddle $\theta_{n-1}$ as it has been described above for the vacua that lie within $[0,\pi/2)$. In addition, $\theta_n$ connects with the lowest vacuum in $[\pi/2,\pi)$ which is the sink $\theta_{n+1}$ at the $\phi=0$ semicircle; the solutions are similar to the ones that we have found in the $SU(2)$ $n=1$ model in section five. The sink $\theta_{n+1}$ also connects to the source $\theta_{n+3}$ and the saddle $\theta_{n+2}$ as required by the reflection symmetry. It is straightforward to repeat this analysis for the $n=4\ell+3$ case. In figures 1 and 2, we confirm our results by a numerical computation for the solutions of equation   of the $SU(2)$ $n=3, 5$ models that lie in the complement of the maximal torus of $SU(2)$. We remark that it is possible to interpolate amongst the vacua of $SU(2)$ model with $n\geq 5$ in a way different from the one proposed above. However, the numerical calculation for $n=5$ supports the case that has been presented. $$\vbox{
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            \put 265 586 {\leftarrow}
            \put 283 569 {\rightarrow}
            \put 134 479 {\theta_1}
            \put 134 514 {\theta_2}
            \put 419 531 {\theta_3}
            \put 419 565 {\theta_4}
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            \put 134 618 {\theta_6}
            \put 419 635 {\theta_7}
            \put 419 669 {\theta_8}
            \put 275 576 {\times}
        \put 130 682 {\pi}
            \put 122 575 {\pi/2}
            \put 146 459 {0}
        \put 408 459 {\pi} 
            \put 271 455 {\pi/2} 
            \put  99 568 {\uparrow}
            \put 268 438 {\rightarrow}
            \put 100 584 {\theta}
            \put 284 436 {\phi}
            \endlabels
            \epsfxsize=.65\hsize
            \centerline{\epsfbox{Fig3.eps}}}$$ -

[**Fig. 3 :**]{} The solutions of the ${\rm SU(2)}$ $n=4$ model

Next we turn to examine the $n$ even case and observe that $({\pi\over2}, {\pi\over2})$ is a fixed point of  but [*not*]{} a vacuum of the theory; $({\pi\over2}, {\pi\over2})$ is source for $n=4 \ell$ and a sink for $n=4\ell+2$. For $n=4\ell+2$, the sink $\theta_{n-1}$ connects to the saddle $\theta_{n-2}$ and to the source $\theta_{n-3}$ in the usual way. Both the sink $\theta_{n-1}$ and the saddle $\theta_n$ connect to the $({\pi\over2}, {\pi\over2})$ fixed point. Using the antipodal map, we can easily determine the behaviour of the solutions associated to the vacua $\theta_{n+1}$ and $\theta_{n+2}$ from the solutions associated to the vacua $\theta_n$ and $\theta_{n-1}$. We remark that for $n=2$, we recover the behaviour we have found in section six. It is straightforward to repeat this analysis for the $n=4\ell$ case. In figure 3, we confirm our results with a numerical computation for the solutions of   of the $SU(2)$ $n=4$ model that lie in the complement of the maximal torus of $SU(2)$.

The $G$-models
==============

In the previous sections we have investigated the soliton solutions of sigma models with target space the groups $SO(2)$ and $SU(2)$. In this section, we shall explore some of the properties of the soliton solutions of $G$-models for $G$ any semisimple compact Lie group. The vacua of any $G$-model can be arranged to lie on a maximal torus of the group $G$. To show this, we first choose, without loss of generality, $\kappa$ in a Cartan subalgebra of $G$ as in section two, i.e. $$\kappa=\kappa^m t_m\ ,
\eqn\gone$$ where $t_m$ is an orthonormal basis of the Cartan subalgebra and $\kappa^m$ are the components of $\kappa$ ($\kappa^m\in \bR$). Now suppose that there is a vacuum $k_0$ that does not lie on this maximal torus of $G$, then there is a $h_0\in G$ such that $k'_0=h_0 k_0 h_0^{-1}$ and $k_0'$ is in the maximal torus. Using the invariance of the equation for the vacua under conjugation,  can be written as $$2h^{-1}_0\kappa h_0+\sum _A t_A{\rm tr}\big(t_A(k'_0)^n\big)=0\ ,
\eqn\goneaa$$ where $\kappa$ is as in . Now, this equation has solutions if and only if $h_0\kappa h_0^{-1}$ is again in the Cartan subalgebra of $G$. It follows then that $h_0$ is an element of the Weyl group of $G$ in which case $k_0$ is in the maximal torus. Therefore all vacua of the $G$-models are on the maximal torus of $G$. This allows us to set $$k=\exp\big({\theta^m t_m}\big)
\eqn\gtwo$$ and rewrite the equation for the vacua as $$2\kappa_m+{\rm tr}\big(t_m \exp({n\theta^r t_r})\big)=0\ .
\eqn\gthree$$ Observe that the second term in the equation for the vacua is the derivative of the character $\chi(\theta)={\rm tr}\, k$ of $G$ with respect to the co-ordinates of the maximal torus.

As in the $SU(2)$ model, the $G$-model also has two classes of solitons. (i) The static solitons that lie on the maximal torus of $G$. (ii) The time-dependent solitons, apart from their asymptotic values, lie in the complement of the maximal torus of $G$. In the former case, the equations   can be written as $${d \over dx}\theta^m(x)=\mp {m\over2} \bigg(2\kappa^m + {\rm tr}
\big(t^m \exp(n\theta^r t_r)\big)\bigg)\ .
\eqn\gfour$$ A class of solutions of these equations can be found by setting all but one angle, say $\theta^1$, to their vacuum values. The solitons obtained in this way are precisely those that are embeddings of the $SO(2)$ solitons along the direction of $\theta^1$. There are as many independent directions for embedding $SO(2)$ solitons in the $G$-model as the rank of the group $G$. Apart from these solitons, there are other static solitons particular to the $G$-model ($\rm{rank}\,{\cal
L}(G)>1$) for which more than one of the angles $\theta^m$ are not set to their vacuum values, and the equations  do not separate to a sum of $SO(2)$ ones. To see this, let us consider the $SU(3)$ model. The maximal torus in this case is $T^2$ and the theory has four vacua for some choice of $\kappa$. The equations  are similar to the standard Morse flow of the height function $f$ of $T^2$. Apart from the Morse flows of $f$ which one gets by embedding the Morse flows of $S^1$ along the two homology cycles of $T^2$ and which are similar to the embedding of $SO(2)$ solitons in the $SU(3)$ model, one obtains additional flows, for example an one-parameter family of flows from the absolute maximum to the absolute minimum of $f$.

The investigation of the time-dependent solitons that lie in the complement of the maximal torus of $G$ is more involved because, apart from the Bogomol’nyi equations along the maximal torus, we should also solve the Bogomol’nyi equations along the root directions of $G$. Despite this, some of them can be obtained from embedding the $SU(2)$ time-dependent solitons into the $G$-model up to a sigma model transformation  of the $G$ model. This is done by embedding the $SU(2)$ group into the $G$ group up to a conjugation. This group theoretical problem has been investigated at the Lie algebra level in \[\] and it was found that there is at least one embedding of $SU(2)$ in $G$ for every positive root $\alpha$ of $G$. For such an embedding the coupling constant $\kappa$ of the $G$-model is $$\kappa^m=\tilde\kappa \alpha^m \ ,
\eqn\gtwo$$ where $\tilde\kappa$ is a real number and $|\tilde \kappa|
<1/|\alpha|^2$, [i.e.]{} $\kappa$ is along a direction in the Cartan subalgebra of $G$ labeled by the root $\alpha$. The generators of the $SU(2)$ subgroup of $G$ are $$i  \tilde h_\alpha, \qquad i(\tilde E_\alpha+\tilde E_{-\alpha})\ , 
\qquad \tilde E_\alpha-\tilde E_{-\alpha} \ ,
\eqn\gthree$$ where $\{\tilde h_\alpha ,\tilde E_\alpha, \tilde E_{-\alpha}\}$ are the generators of ${\cal L}(G)$ in the Chevalley basis. It is clear from  that all these embeddings do not describe the most general soliton solutions of the $G$-model. For this, one should solve the Bogomol’nyi equations of the $G$-model for $\kappa$ along a generic direction in the Cartan subalgebra of $G$.

We expect that apart from the one-soliton solutions that we have discussed so far, the $G$-models may have multi-soliton solutions as well. Indeed for $G$-models with $\kappa=0$, we can simply embed the multi-soliton solutions of the sine-Gordon theory (see for example \[\] and references within.). However, since the multi-soliton solutions of the sine-Gordon theory are not solutions of the Bogomol’nyi equations, the same applies for embedded multi-soliton solutions in the $G$-model.

Concluding remarks and summary
==============================

Quantum mechanically, some solitons may decay to other solitons that carry the same topological charge but different Noether charge by emitting radiation. This is because, for $G$-models for which $G$ is a semisimple group, the fundamental states of the theory as well as its solitons are charged with respect to the [*same*]{} Noether charge $Q_N$. Note that the solitons, in addition to the Noether charge $Q_N$, carry topological charge $Q_T$. From charge conservation it follows that a soliton with topological charge $Q_T$ and with Noether charge $Q_N$ may decay to an other soliton with the same topological charge but with Noether charge $Q_N'$ and $|Q_N'|<|Q_N|$, and some fundamental states with Noether charge $Q_N-Q'_N$. Energy conservation though seems to rule out such a process because the fundamental states do not saturate the bound ($\kappa\not=1$). However if for some unknown mechanism to us such a process is allowed the stable soliton configurations are those that have the least mass for given topological charge. An example of such a configuration is the soliton  of the $SU(2)$ $n=1$ model. Moreover, as it is well known, the coupling constant of $G$-models, $G$ semisimple, is quantised due to the presence of the torsion term and therefore it may be more appropriate to develop a non-perturbative method, similar to that developed for the Wess-Zumino-Witten model \[\], to investigate the fundamental states of the theory instead of linearising the theory about a vacuum in the weak coupling limit. Another related issue is the quantisation of the charge $Q$ of the solitons. Since $Q$ is not purely topological, the quantisation of $Q$ does not follow from classical considerations; a quantisation of the moduli space suggests though that $Q$ should be quantised provided that the moduli coordinate $\psi$ is periodic which is the case whenever the orbits of the vector field $X$ in $G$ are periodic. We also remark that even if $Q$ is quantised the usual stability argument for BPS monopoles does not apply to this case because the bound for the energy is in terms of one charge rather than two, which are necessary to establish the stability for the solitons using the triangular inequality.

To summarise, we have investigated the soliton solutions of (1,1)-supersymmetric massive sigma models with torsion and target space a group $G$ for a class of scalar potentials characterised by a coupling constant $\kappa$ and an integer $n$. These solitons are solutions of Bogomol’nyi equations which arise from the saturation of a bound for the energy of these models in terms of a charge $Q$ that appears as a central charge in the (1,1)-supersymmetry algebra. The charge $Q$ is the sum of a Noether charge $Q_N$ and a topological charge $Q_T$. The $G=SO(2)$ model is the simplest of the $G$-models and its solitons can be easily computed; the $SO(2)$ model with $\kappa=0$ is the supersymmetric sine-Gordon theory. For $G$ a semisimple group there are two classes of solitons to consider, one is a set of static solitons that lie on a maximal torus of $G$ and the other is a set of time-dependent solitons that, apart from their asymptotic values, lie in the complement of a maximal torus of $G$. For $G=SU(2)$, we have found all static solitons as embeddings of the corresponding $SO(2)$ solitons. In addition, we have explicitly computed the time-dependent solitons of the $n=1$ model, and we have shown that all solitons of the $n=2$ model are static. We have also presented the qualitative properties of the time-dependent solitons of the $SU(2)$ $n>2$ models and confirmed our results with a numerical calculation for the $n=3,4,5$ ones. For sigma models with target space a semisimple group $G$, some of their solitons can be obtained from embedding the solitons of the $SO(2)$ model and the solitons of the $SU(2)$ model with the corresponding value of $n$.

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[**Acknowledgments:**]{} G.P. is supported by a University Research Fellowship from the Royal Society. We would like to thank A. Sornborger and P.K. Townsend for helpful comments.
