Magnetic translation groups in an n-dimensional torus

TL;DR

This paper explores the topology and symmetry of a charged particle in a magnetic field on an n-dimensional torus, generalizing the concept of magnetic translation groups and their representations to higher dimensions.

Contribution

It defines the magnetic translation group on an n-torus as a central extension of a cyclic group product and classifies its irreducible unitary representations, extending noncommutative torus concepts.

Findings

Magnetic translation group is a central extension of cyclic groups and tori.

Constructed and classified irreducible representations on a three-torus.

Generalized the noncommutative torus concept to higher dimensions.

Abstract

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus.