Higher dimensional quantum Hall effect and the analog of $W_\infty$-algebra

Abstract

We show that Abelian and nonabelian gauge transformations are the analog of $W_\infty$ transformations for higher dimensional quantum Hall effect. The commutator anomaly and the extended algebra of such transformations on the edge modes of a droplet are obtained by purely topological arguments, basically utilizing the two-cocycle in the descent procedure for anomalies and using the fact that there is anomaly cancellation between the bulk and boundary actions. The method relies on the fact that bulk actions are easily constructed in general using the Dolbeault index theorem. The resulting algebras are shown to agree with explicit edge mode calculations for cases where they are available. We also comment on the similarities and differences in the nature of these transformations between two and higher dimensions.