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 