-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 