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

TL;DR

This paper demonstrates an exact equivalence between the O(3) sigma model with a Hopf term and a higher spin theory, revealing a deep connection between topological terms and spin representations.

Contribution

It introduces a local CP_1 model with gauge fields that precisely maps to the sigma model with Hopf term, establishing a link to higher spin theories at specific coupling values.

Findings

Exact equivalence between sigma model with Hopf term and higher spin theory.

Identification of a fixed point where the theories coincide.

Conjecture of a fixed point in the spin-s theory near the sigma model.

Abstract

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.