Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT

TL;DR

This paper compares two methods of computing AdS/CFT correlators, highlights the importance of operator renormalization, and discusses the inequivalence of dS/CFT analogues, also exploring the analytic continuation of wave functions.

Contribution

It clarifies the equivalence of boundary and bulk dictionary approaches in AdS/CFT at the path integral level, emphasizing renormalization, and examines the differences in dS/CFT, including wave function continuation.

Findings

Bulk operator renormalization is crucial for dictionary agreement.

AdS/CFT boundary and bulk approaches are equivalent with proper renormalization.

dS/CFT dictionaries are fundamentally inequivalent.

Abstract

Dual AdS/CFT correlators can be computed in two ways: differentiate the bulk partition function with respect to boundary conditions, or extrapolate bulk correlation functions to the boundary. These dictionaries were conjectured to be equivalent by Banks, Douglas, Horowitz, and Martinec. We revisit this question at the level of bulk path integrals, showing that agreement in the presence of interactions requires careful treatment of the renormalization of bulk composite operators. By contrast, we emphasize that proposed dS/CFT analogues of the two dictionaries are inequivalent. Next, we show quite generally that the wave function for Euclidean AdS analytically continues to the dS wave function with Euclidean initial conditions. Most of our arguments consider interacting fields on a fixed background, but in a final section we discuss the inclusion of bulk dynamical gravity.