  We write down a local $CP_1$ model involving two gauge fields, which is
exactly equivalent to the O(3) $\sigma$ model with the Hopf term. We impose the
$CP_1$ constraint by using the gaussian representation of the delta function.
For the coefficient of the Hopf term, $\theta = {\pi \over 2s}$, 2s being an
integer, we show that the resulting model is exactly equivalent to an
interacting theory of spin-$s$ fields. Thus we conjecture that there should be
a fixed point in the spin-$s$ theory near which it is exactly equal to the
$\sigma$ model.
