Holographic renormalization for irrelevant operators and multi-trace counterterms

TL;DR

This work extends holographic renormalization to settings with sources for irrelevant operators ($ riangle > d$) by developing a perturbative framework in the sources. It identifies two main new structures: pseudo-non-local divergences that are local in the cutoff data but non-local in the source, and genuine non-local divergences that require multi-trace counterterms involving the conjugate momentum $ ext{Π}_r$, thereby refining the gravity–field-theory dictionary. The analysis is carried out in two toy models: an interacting scalar in fixed AdS and a free scalar in a general AlAdS background with backreaction, showing that backreaction is essential for a consistent renormalization and that multi-trace counterterms arise at higher orders. The results support a renormalizable dual theory under parametrically small irrelevant deformations and suggest a general framework where counterterms are local in $( ext{Φ}, ext{Π}_r)$ rather than in $ ext{Φ}$ alone, with significant implications for non-AlAdS holography and operator mixing.

Abstract

We investigate the structure of holographic renormalization in the presence of sources for irrelevant operators. By working perturbatively in the sources we avoid issues related to the non-renormalizability of the dual field theory. We find new classes of divergences which appear to be non-local on the gravity side. However in all cases a systematic renormalization procedure exists involving either standard local counterterms or new counterterms which may be interpreted as multi-trace counterterms in the field theory. The multi-trace counterterms reflect a more intricate relation between sources and the asymptotics of bulk fields.