$D$-algebra structure of topological insulators

In the quantum Hall effect, the density operators at different wave vectors generally do not commute and give rise to the Girvin-MacDonald-Plazmann (GMP) algebra, with important consequences such as ground-state center-of-mass degeneracy at fractional filling fraction, and $W_{1+\infty }$ symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher-dimensional topological insulators involves the concept of a $D$ commutator. For insulators in even-dimensional space, the $D$ commutator is isotropic and closes, and its structure factors are proportional to the $D/2$ Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one fewer dimension), and does not contain the Chern-Simons $\theta$ form. This algebraic structure paves the way towards the identification of fractional topological insulators through the counting of their excitations. The possible relation to $D$-dimensional volume-preserving diffeomorphisms and parallel transport of extended objects is also discussed.