$$ simplifies both the Bogomol'nyi equations \oneb\ and the 
equation for the vacua  \fourb\ of the
model. In particular, we shall show in chapter eight that it can be
arranged for the vacua to lie in the
maximal torus of the group $G$.

 
The solutions of the Bogomol'nyi equations may depend on several
moduli parameters.  In that case, as
for BPS monopoles [\manton], one can define a metric on the moduli
space of solutions as
follows:
$$
ds^2={1\over2}\int dx\, g_{ij} d\phi^i d\phi^j\ ,
\eqn\sevenb
$$
where the differential on $$ denotes variation with respect to the
moduli parameters. The soliton
solutions of the Bogomol'nyi equations \oneb\ preserve $1/2$ of the 
supersymmetry and therefore their
effective theory is an $N=1$ supersymmetric one-dimensional sigma
model with target space the moduli space and
metric the moduli metric \sevenb. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Chapter 2%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The SO(2) model}

The simplest of the $G$-models is the one with the group $SO(2)$ as
target space.
Since the target space is one dimensional, the antisymmetric tensor
coupling $b$ is identically zero.  A
parameterisation of the group manifold
$SO(2)$ is 
$$
k=\pmatrix{\cos\theta & \quad -\sin\theta\cr \sin\theta & \quad \cos\theta}
\eqn\onec
$$
and a basis in $[L]{}(SO(2))$ is
$$
t_1=\pmatrix{0&\quad -1\cr 1&\quad 0}\ 