  We construct a generalization of the quantum Hall effect where particles move
in an eight dimensional space under an SO(8) gauge field. The underlying
mathematics of this particle liquid is that of the last normed division
algebra, the octonions. Two fundamentally different liquids with distinct
configurations spaces can be constructed, depending on whether the particles
carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20
dimensional manifold of with an internal component of SO(7) holonomy, whereas
the second liquid lives on a 14 dimensional manifold with an internal component
of $G_2$ holonomy.
