Quantum Hall effect in higher dimensions

Following recent work on the quantum Hall effect on S⁴, we solve the Landau problem on the complex projective spaces CP ^k and discuss quantum Hall states for such spaces. Unlike the case of S⁴, a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of CP ² in some detail considering both Abelian and nonAbelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of CP ³ is related to the case of S⁴.