On Holography in General Background and the Boundary Effective Action from AdS to dS

TL;DR

This paper develops a holographic framework for quantum fields on arbitrary backgrounds, deriving boundary actions, propagators, and one-loop corrections in AdS and dS spaces, with applications to CFT operator data and cosmological correlators.

Contribution

It introduces a holographic basis for scalar fields on arbitrary backgrounds, deriving boundary actions, propagators, and one-loop effective actions in AdS and dS, advancing holographic and cosmological analyses.

Findings

Derived exact flux formulas in warped backgrounds.

Computed boundary effective actions and anomalous dimensions in AdS.

Analyzed radiative effects and operator mixing in boundary CFTs.

Abstract

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.