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We formulate the Landau problem in the context of the noncommutative analog of a surface of constant negative curvature, that is AdS 2 surface, and obtain the spectrum and contrast the same with the Landau levels one finds in the case of the commutative AdS 2 space.

The fundamental groups of the configuration spaces for the O(3) nonlinear σ-model on the compact genus g surfaces [Formula: see text] and on the connected sums [Formula: see text] are known for any soliton number N. So are the braid for N spinless particles on these manifolds. The representations of these groups govern the possible statistics of solitons and particles. We show that when spin and creation/annihilation processes are introduced, the fundamental groups for the particles are the same as the corresponding σ-model groups. These fundamental groups incorporate the spin-statistics connection and are of greater physical relevance than the standard braid groups.

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