Entanglement for Quantum Hall states and a Generalized Chern-Simons Form

We analyze some features of the entanglement entropy for an integer quantum Hall state ($\nu =1$) in comparison with ideas from relativistic field theory and noncommutative geometry. The spectrum of the modular operator, for a restricted class of states, is shown to be similar to the case of field theory or a type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ von Neumann algebra. We present arguments that the main part of the dependence of the entanglement entropy on background fields and geometric data such as the spin connection is given by a generalized Chern-Simons form. Implications of this result for bringing together ideas of noncommutative geometry, entropy and gravity are briefly commented upon.