Issues on 3D Noncommutative Electromagnetic Duality

TL;DR

This paper investigates the extension of 3D electromagnetic duality to noncommutative space-time using Seiberg-Witten maps, revealing limitations of duality at higher orders and providing simplified formulations of the mapped Lagrangian.

Contribution

It extends 3D NC electromagnetic duality to second order, analyzes its breakdown at third order, and simplifies the Seiberg-Witten mapped Lagrangian to all orders.

Findings

Duality interchanges theta with its Hodge dual times the gauge coupling at second order.

Duality fails at third order unless slowly varying fields limit is assumed.

A simplified expression for the Seiberg-Witten mapped Lagrangian is derived for 3D NC theories.

Abstract

We extend the ordinary 3D electromagnetic duality to the noncommutative (NC) space-time through a Seiberg-Witten map to second order in the noncommutativity parameter (theta), defining a new scalar field model. There are similarities with the 4D NC duality, these are exploited to clarify properties of both cases. Up to second order in theta, we find that duality interchanges the 2-form theta with its 1-form Hodge dual *theta times the gauge coupling constant, i.e., theta --> *theta g^2 (similar to the 4D NC electromagnetic duality). We directly prove that this property is false in the third order expansion in both 3D and 4D space-times, unless the slowly varying fields limit is imposed. Outside this limit, starting from the third order expansion, theta cannot be rescaled to attain an S-duality. In addition to possible applications on effective models, the 3D space-time is useful for studying general properties of NC theories. In particular, in this dimension, we deduce an expression that significantly simplifies the Seiberg-Witten mapped Lagrangian to all orders in theta.