  We study quantum fields on an arbitrary, rigid background with boundary. We
derive the action for a scalar in the holographic basis that separates the
boundary and bulk degrees of freedom. A relation between Dirichlet and Neumann
propagators valid for any background is obtained from this holographic action.
As a simple application, we derive an exact formula for the flux of bulk modes
emitted from the boundary in a warped background. We also derive a formula for
the Casimir pressure on a $(d-1)$-brane depending only on the boundary-to-bulk
propagators, and apply it in AdS. Turning on couplings and using the
holographic basis, we evaluate the one-loop boundary effective action in AdS by
means of the heat kernel expansion. We extract anomalous dimensions of single
and double trace CFT operators generated by loops of heavy scalars and
nonabelian vectors, up to third order in the large squared mass expansion. From
the boundary heat kernel coefficients we identify CFT operator mixing and
corrections to OPE data, in addition to the radiative generation of local
operators. We integrate out nonabelian vector fluctuations in AdS$_{4,5,6}$ and
obtain the associated holographic Yang-Mills $\beta$ functions. Turning to the
expanding patch of dS, following recent proposals, we provide a boundary
effective action generating the perturbative cosmological correlators using
analytical continuation from dS to EAdS. We obtain the "cosmological" heat
kernel coefficients in the scalar case and work out the divergent part of the
dS$_4$ effective action which renormalizes the cosmological correlators. We
find that bulk masses and wavefunction can logarithmically run as a result of
the dS$_4$ curvature, and that operators on the late time boundary are
radiatively generated. More developments are needed to extract all one-loop
information from the cosmological effective action.
