Conformal actions in any dimension

In standard d-dimensional conformal gauge theory, the volume element scales with weight d, so that a given form of the action is scale invariant only in a single dimension. However, the new biconformal gauging of the conformal group has a scale invariant volume form, permitting a single form of the action to be invariant for any value of d. We show that the simplest scale invariant action is linear in the curvatures, and the resulting field equations are satisfied in a torsion-free biconformal space if and only if the space is in 1-1 correspondence with a d-dim scale invariant geometry in which the Weyl vector possesses a linear momentum dependence. Further, we display curvature-quadratic actions which are conformally invariant for every d.