Double Trace Flows and Holographic RG in dS/CFT correspondence

TL;DR

This work investigates how holographic RG flows in de Sitter space (dS) encode time evolution of the bulk, focusing on double- and triple-trace deformations in the putative dS/CFT dual. The authors derive the beta function for double-trace couplings from holography, showing it takes the universal form $\beta(\lambda) = -2\nu \lambda + 2\nu b_{dS} \lambda^2$ with a coefficient $b_{dS}$ fixed by bulk data; the RG scale is identified with $\mu \sim (-iT)$, where $T$ is the bulk time in inflationary coordinates. In $dS_4$ there exists a normalization making both the boundary correlators real and the beta-function coefficients real, yielding an IR fixed point at negative coupling and linking to the Sp(N) vector model/Vasiliev gravity correspondence; for even $d$ in general, reality of all correlators and beta-function coefficients cannot be simultaneously achieved without introducing $n$-dependent phases in higher-point functions. The paper extends the holographic dictionary to dS by computing two- and higher-point functions under standard and alternative quantization, derives the double-trace and triple-trace beta-functions from the bulk Hamilton–Jacobi framework, and clarifies how bulk time evolution translates into RG flow in the non-unitary boundary theory, with explicit structure for triple and higher trace couplings. Overall, the results illuminate how unitary bulk dynamics imprint constraints on a non-unitary boundary CFT and demonstrate concrete holographic realizations, including the role of operator normalizations and phase structures in even spacetime dimensions.

Abstract

If there is a dS/CFT correspondence, time evolution in the bulk should translate to RG flows in the dual euclidean field theory. Consequently, although the dual field is expected to be non-unitary, its RG flows will carry an imprint of the unitary time evolution in the bulk. In this note we examine the prediction of holographic RG in de Sitter space for the flow of double and triple trace couplings in any proposed dual. We show quite generally that the correct form of the field theory beta functions for the double trace couplings is obtained from holography, provided one identifies the scale of the field theory with (i|T|) where T is the `time' in conformal coordinates. For dS(4), we find that with an appropriate choice of operator normalization, it is possible to have real n-point correlation functions as well as beta functions with real coefficients. This choice leads to an RG flow with an IR fixed point at negative coupling unlike in a unitary theory where the IR fixed point is at positive coupling. The proposed correspondence of Sp(N) vector models with de Sitter Vasiliev gravity provides a specific example of such a phenomenon. For dS(d+1) with even d, however, we find that no choice of operator normalization exists which ensures reality of coefficients of the beta-functions as well as absence of n-dependent phases for various n-point functions, as long as one assumes real coupling constants in the bulk Lagrangian.