determine $M,N$ from the first two equations in ; these equations will be studied in the next three chapters for various values of $n$.

It is convenient for reasons that will become apparent later to parameterise $SU(2)$ in angular coordinates as follows: $$\eqalign{
M&=\cos\theta
\cr
N&=\sin\theta \cos\phi
\cr
U&=\sin\theta  \sin\phi \cos\psi
\cr 
W&=\sin\theta \sin\phi \sin\psi\ ,}
\eqn\nineda$$ where $0\leq \theta, \phi<\pi$ and $0\leq \psi<2\pi$. In this parameterisation, the equation  for the vacua can be written as $$\eqalign{
\sin(n\theta)&=\pm \kappa
\cr
\cos\phi&=\pm 1\ . }
\eqn\nfived$$ Note that the vacua of the theory lie in $\phi=0$ and $\phi=\pi$ semi-circle subspaces of $SU(2)$, which are joined at $(\theta, \phi)=(0,0)$ and $(\theta, 
\phi)=(0,\pi)$, and at $(\theta,
\phi)=(\pi,0)$ and $(\theta, \phi)=(\pi,\pi)$ to a circle in $SU(2)$; in fact this circle is a maximal torus of $SU(2)$. Solving  for $0\leq \kappa <1$, we find that the $2n$ vacua of the model are $$\big(\theta, \phi\big)=\cases{\bigg({1\over
n}(\pm\theta_0+{1\mp1\over2}\pi+2\pi \tilde \ell), 0\bigg)\ ,
\qquad\qquad 0\leq\tilde
\ell\leq \big[{n-1\over2}\big]
\cr
\bigg({1\over n}\big(\pm\theta_0+{1\mp1\over2}\pi+2\pi (\tilde 
\ell+{1\over2})\big), \pi\bigg)\ ,\qquad 0\leq\tilde
\ell<\big[{n\over2}\big]\ ,}
\eqn\vacthree$$ where $0\leq \theta_0<{\pi\over2}$ and $\sin \theta_0=\kappa$. For $\kappa=0$, $\theta_0$ vanishes and the expression for the vacua becomes $$\eqalign{
(\theta, \phi)&=\big({\ell \over n}\pi, 0\big)
\cr
(\theta, \phi)&=\big({\ell+1 \over n}\pi, \pi\big)\ , }
