Is the $O(3)~σ$ Model with the Hopf Term Exactly Equivalent to a Higher Spin Theory?

Abstract

We write down a local $CP_1$ model involving two gauge fields, which is exactly equivalent to the O(3) $σ$ 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, $θ= {π\over 2s}$, 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 $σ$ model.