Interacting quantum topologies and the quantum Hall effect

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by A(theta)(R-2) circle times A(theta)(R-2) (theta = -4/eB), the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding center coordinates. We argue that A(theta)(R-2) 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 A(theta)(R-2) to electromagnetic fields which are described by the Abelian commutative gauge group G(c)(U(1)), i.e. gauge fields based on A(0)(R-2). 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 A(theta)' with different theta' appear. Two-particle wave functions in this approach are based on A(theta)(R-2) circle times A(theta)(R-2). We find that the full symmetry group in IQHE, which is the semidirect product SO(2) x G(c)(U(1)) acts on this tensor product using the twisted coproduct Delta(theta). 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.