$W_{1+\infty}$ Dynamics of Edge Excitations in the Quantum Hall Effect
  Quantum Hall universality classes can be classified by $W_{1+\infty}$ symmetry. We show that this symmetry also governs the dynamics of quantum edge excitations. The Hamiltonian of interacting electrons in the fully-filled first Landau level is expressed in terms of $W_{1+\infty}$ generators. The spectra for both the Coulomb and generic short-range interactions are thus found algebraically. We prove the one-dimensional bosonization of edge excitations in the limit of large number of particles. Moreover, the subleading corrections are given by the higher-spin $W_{1+\infty}$ generators, which measure the radial fluctuations of the electron density. The resulting spectrum for the Coulomb interaction contains a logarithmic enhancement, in agreement with experimental observations. The spectrum for generic short-range interactions is subleading and reproduces the classical capillary frequencies. These results are also extended to fractional filling by using symmetry arguments. 
A. Cappelli, C. A. Trugenberger and G. R. Zemba