Conformal invariance and the regularised one-loop effective action

A unique family G n-1 of conformal invariants, polynomial in the extrinsic curvature of a hypersurface embedded in an n-dimensional Riemannian manifold M, is constructed. In the quantum theory of a conformally coupled scalar field on a manifold M with boundary delta M, the counterterm that regularises the one-loop effective action contains members of G n-1 integrated over delta M. The counterterms through n=4 are given explicitly and a derivation of the n=4 boundary contribution is given based on a flat-space result.