Soliton operators in the quantum equivalence of the $CP_1$ and $O(3)-\sigma$ models

TL;DR

This paper explores the quantum equivalence between the $O(3)$ sigma model and the $CP_1$ model in 3D, focusing on canonical quantization, functional determinants, and soliton operators to verify their deep connection.

Contribution

It demonstrates the quantum equivalence in all topological sectors using canonical and path integral methods, and constructs soliton operators across models.

Findings

Canonical quantization is free of ordering ambiguities.

Functional determinants verify phase-space partition function equivalence.

Explicit soliton operator construction shows the models' deep connection.

Abstract

We discuss some interesting aspects of the well known quantum equivalence between the $O(3)-\sigma$ and $CP_1$ models in $3D$, working in the canonical and in the path integral formulations. We show first that the canonical quantization in the hamiltonian formulation is free of ordering ambiguities for both models. We use the canonical map between the fields and momenta of the two models and compute the relevant functional determinant to verify the equivalence between the phase-space partition functions and the quantum equivalence in all the topological sectors. We also use the explicit form of the map to construct the soliton operator of the $O(3)-\sigma$ model starting from the representation of the operator in the $CP_1$ model, and discuss their properties