The Polynomial Formulation of the U(1) Non-Linear Sigma-Model in 2 Dimensions

TL;DR

This paper presents a polynomial formulation of the U(1) non-linear sigma-model in two dimensions, revealing dualities, vortex roles, and connections to the Sine-Gordon model, with a focus on O(2) invariance.

Contribution

It introduces a first-order polynomial formulation using 1-form and 0-form fields, providing new insights into vortex dynamics and dualities in the model.

Findings

Vortices are dual to spin variables in the partition function.

The Sine-Gordon model naturally emerges from the formulation.

Strings of vortices can be incorporated into the framework.

Abstract

We investigate some properties of a first-order polynomial formulation of the U(1) non-linear sigma-model in two Euclidean dimensions. The variables in this description are a 1-form field plus a 0-form Lagrange multiplier field. The usual spin variables are non-local functions of the new fields. As this construction incorporates O(2) invariance ab initio, only O(2)-invariant correlation functions (the only non-vanishing ones in the model) can be constructed. We show that the vortices play a dual role to the spin variables in the partition function. The equivalent Sine-Gordon description is obtained in a natural way, when one integrates out the 1-form field to get an effective partition function for the Lagrange multiplier. We also show how to introduce strings of vortices within this formulation.