Matrix model description of Laughlin Hall states

Abstract

We analyse Susskind’s proposal of applying the non-commutative Chern–Simonstheory to the quantum Hall effect. We study the corresponding regularized matrixChern–Simons theory introduced by Polychronakos. We use holomorphic quantization andperform a change of matrix variables that solves the Gauss law constraint. Theremaining physical degrees of freedom are the complex eigenvalues that can beinterpreted as the coordinates of electrons in the lowest Landau level with Laughlin’swavefunction. At the same time, a statistical interaction is generated among the electronsthat is necessary to stabilize the ground state. The stability conditions can beexpressed as the highest-weight conditions for the representations of the W∞ algebra in the matrix theory. This symmetry provides a coordinate-independentcharacterization of the incompressible quantum Hall states.