Revisiting AdS/CFT at a finite radial cut-off

We revisit AdS/CFT at a finite radial cut-off, specifically in the context of double trace perturbations, On=Ox∂2nOx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{O}}_n=\mathcal{O}(x){\left({\partial}^2\right)}^n\mathcal{O}(x) $$\end{document}, with arbitrary powers n. As well-known, the standard GKPW prescription, applied to a finite radial cut-off, leads to contact terms in correlators. de Haro et al. [1] introduced bulk counterterms to remove these. This prescription, however, yields additional terms in the correlator corresponding to spurious double trace deformations. Further, if we view the GKPW prescription coupled with the prescription in [1], in terms of a boundary wavefunction, we find that it is incompatible with radial Schrödinger evolution (in the spirit of holographic Wilsonian RG). We consider a more general wavefunction satisfying the Schrödinger equation, and find that generically such wavefunctions generate both (a) double trace deformations and (b) contact terms. However, we find that there exist special choices of these wavefunctions, amounting to a new AdS/CFT prescription at a finite cut-off, so that both (a) and (b) are removed and we obtain a pure power law behaviour for the correlator. We compare these special wavefunctions with a specific RG scheme in field theory. We give a geometric interpretation of these wavefunctions; these correspond to some specific smearing of boundary points in the Witten diagrams. We present a comprehensive calculation of exact double-trace beta-functions for all couplings On\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{O}}_n $$\end{document} and match with a holographic computation using the method described above. The matching works with a mapping between the field theory and bulk couplings; such a map is highly constrained because the beta-functions are quadratic and exact on both sides. Our discussions include a generalization of the standard double-trace Wilson-Fisher flow to the space of the infinite number of couplings.