Seiberg-Witten map and Galilean symmetry violation in a non-commutative planar system

TL;DR

This paper constructs an effective gauge-invariant theory for a non-commutative 2+1D Schrödinger system, revealing Galilean symmetry violation and analyzing Hall conductivity within this non-commutative framework.

Contribution

It introduces a first-order Seiberg-Witten map-based effective theory for non-commutative Schrödinger fields with background gauge fields, highlighting symmetry violations.

Findings

Galilean boost symmetry is violated in the model.

The effective theory resembles the usual Schrödinger action with noncommutative interactions.

Hall conductivity is analyzed within the non-commutative setting.

Abstract

An effective U(1) gauge invariant theory is constructed for a non-commutative Schrodinger field coupled to a background U(1)_{\star} gauge field in 2+1-dimensions using first order Seiberg-Witten map. We show that this effective theory can be cast in the form of usual Schrodinger action with interaction terms of noncommutative origin provided the gauge field is of ``background'' type with constant magnetic field. The Galilean symmetry is investigated and a violation is found in the boost sector. We also consider the problem of Hall conductivity in this framework.