Magnetic translation groups in n dimensions

TL;DR

This paper generalizes magnetic translation groups in n dimensions, representing magnetic fields as antisymmetric tensors and deriving formulae for their central extensions, which broadens understanding of gauge symmetries in higher-dimensional quantum systems.

Contribution

It introduces a general framework for magnetic translation groups in n dimensions, modeling magnetic fields as antisymmetric tensors and deriving explicit formulae for their central extensions.

Findings

Magnetic fields can be represented as antisymmetric tensors in n dimensions.

Derived explicit formulae for the central extensions of translation groups.

Provided a unified approach to magnetic translation groups in higher dimensions.

Abstract

Magnetic translation groups are considered as central extensions of the translation group T=Z^n by the group of factors (a~gauge group) U(1). The obtained general formulae allow to consider a magnetic field as an~antisymmetric tensor (of rank 2) and factor systems are determined by a transvection of this tensor with a tensor product t \otimes t'.