Non-commutative Duality: High Spin Fields and $CP^1$ Model with Hopf Term

TL;DR

This paper demonstrates a duality between a non-commutative $CP^1$ model with a Hopf term and an interacting spin-$s$ theory, showing that the duality and topological features are preserved under non-commutativity.

Contribution

It establishes a non-commutative duality using the Seiberg-Witten map, revealing that the spin and topological index remain unaffected by non-commutativity, while the Noether charge depends on it.

Findings

Duality between non-commutative $CP^1$ model and spin-$s$ theory established.

Topological index of solitons is unaffected by non-commutativity.

Noether charge of dual particles depends on the non-commutative parameter $ heta$.

Abstract

We show that the non-commutative $CP^1$ model coupled with Hopf term in 3 dimensions is equivalent to an interacting spin-$s$ theory where the spin $s$ of the dual theory is related to the coefficient of the Hopf term. We use the Seiberg-Witten map in studying this non-commutative duality equivalence, keeping terms to order $\theta$ and show that the spin of the dual theory do not get any $\theta$ dependant corrections. The map between current correlators show that topological index of the solitons in the non-commutative $CP^1$ model is unaffected by $\theta$ where as the Noether charge of the corresponding dual particle do get a $\theta$ dependence. We also show that this dual theory smoothly goes to the limit $\theta\to 0$ giving dual theory in the commutative plane.