  Using noncommutative geometry we do U(1) gauge theory on the permutation
group $S_3$. Unlike usual lattice gauge theories the use of a nonAbelian group
here as spacetime corresponds to a background Riemannian curvature. In this
background we solve spin 0, 1/2 and spin 1 equations of motion, including the
spin 1 or `photon' case in the presence of sources, i.e. a theory of classical
electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs
from the U(1) Maxwell theory in noncommutative geometry), including the moduli
spaces of flat connections. We show that the Yang-Mills action has a simple
form in terms of Wilson loops in the permutation group, and we discuss aspects
of the quantum theory.
