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 configuration 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 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.

• Received 10 June 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.236803