The algebra of observables of planar electrons subject to a constant background magnetic field B is given by [Formula: see text], the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding center coordinates. We argue that [Formula: see text] itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on [Formula: see text] to electromagnetic fields which are described by the Abelian commutative gauge group [Formula: see text], i.e. gauge fields based on [Formula: see text]. This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras [Formula: see text] with different θ′ appear. Two-particle wave functions in this approach are based on [Formula: see text]. We find that the full symmetry group in IQHE, which is the semidirect product [Formula: see text] acts on this tensor product using the twisted coproduct Δθ. Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.
Elastic gauge fields and zero-field three-dimensional quantum Hall effect in hyperhoneycomb lattices
