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A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a contin- uous commutative translation symmetry. We study topology and sym- metry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of transla- tions that leave the gauge field invariant. We show that the MTG in an n-dimensional torus is isomorphic to a central extension of a cyclic group Z1 ×    × Z2l × T m by U(1) with 2l + m = n. We construct and clas- sify irreducible unitary representations of the MTG in a three-torus and apply the representation theory to three examples. We briefly describe a representation theory for a general n-torus. The MTG in an n-torus can be regarded as a generalization of the so-called noncommutative torus.

Magnetic translation groups in an n-dimensional torus and their representations