Electromagnetism and Gauge Theory on the Permutation Group $S_3$

Abstract

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.