Yang-Mills by dimensionally reducing Chern-Simons

We derive the usual first-order form of the Yang-Mills action in arbitrary dimensions by dimensional reduction from a Chern-Simons-like action. The antisymmetric tensor auxiliary field of the first-order action appears as a gauge field for the extra dimensions. The higher-dimensional geometry was introduced in our previous paper by adding dimensions “dual” to spin, as suggested by the superstring’s affine Lie algebra.