,
\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 