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 