Quantum Hall Droplets on Disc and Effective Weiss-Zumino-Witten Action for Edge States

We algebraically analysis the quantum Hall effect of a system of particles living on the disc B1 in the presence of an uniform magnetic field B. For this, we identify the non-compact disc with the coset space SU(1, 1)/U(1). This allows us to use the geometric quantization in order to get the wavefunctions as the Wigner D-functions satisfying a suitable constraint. We show that the corresponding Hamiltonian coincides with the Maass Laplacian. Restricting to the lowest Landau level, we introduce the noncommutative geometry through the star product. Also we discuss the state density behaviour as well as the excitation potential of the quantum Hall droplet. Finally, we show that the edge excitations are described by an effective Weiss-Zumino-Witten action for a strong magnetic field and discuss their nature. Permanent address : Physics Department, Faculty of Sciences, University Ibn Zohr, Agadir, Morocco. e-mail : m−[email protected] e-mail : [email protected] – [email protected]