$SU(2)$ $n=1$ 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.

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\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)\ 