The Quantum Hall Effect on R⁴

Zhang and Hu have formulated an SU(2) quantum Hall system on the four-sphere, with interesting three-dimensional boundary dynamics including gapless states of nonzero helicity. In order to understand the local physics of their model we study the U(1) and SU(2) quantum Hall systems on flat R^4, with flat boundary R^3. In the U(1) case the boundary dynamics is essentially one dimensional. The SU(2) theory can be formulated on R^4 for any isospin I, but in order to obtain a flat boundary theory we must take I \to \infty as in Zhang and Hu. The theory simplifies in the limit, the boundary becoming a collection of one-dimensional systems. We also discuss general constraints on the emergence of gravity from nongravitational field theories.