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

TL;DR

The paper analyzes two bulk–boundary dictionaries in AdS/CFT and dS/CFT, showing their equivalence in AdS/CFT only after carefully renormalizing bulk composite operators, while revealing inequivalence in dS/CFT due to fluctuating boundary modes. It demonstrates that the AdS IR wave function analytically continues to the dS wave function, whereas the two dictionaries lead to different boundary correlators in dS, necessitating a larger dual framework when gravity is included. The work also develops a wave-function-based analytic continuation formalism for both fixed-background QFT and gravity, and discusses perturbative extensions to dynamical gravity with explicit examples and limitations. Overall, the results clarify when and how bulk-to-boundary dictionaries translate between AdS and dS contexts and illuminate the role of renormalization and initial-state choices in holographic dualities.

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.

Figures (6)

• Figure 1: A diagram by which an operator $\phi^3$ can contain an operator $\phi$.
• Figure 2: The contours that produce the dS and Euclidean AdS wave functions in global slicing (left) and flat slicing (right).
• Figure 3: The prescription for computing expectation values of operators at fixed geodesic distance from the boundary in the Euclidean path integral.
• Figure 4: $\ell$-scaling of various diagrams in $\phi^4$ theory. The first and third have four bulk-to-boundary propagators and scale like $\ell^{4\Delta}$, the second has three boundary-to-boundary propagators and scales like $\ell^{6 \Delta}$.
• Figure 5: Diagrams involving an external $\phi^3$ operator in $\phi^4$ theory. The first diagram has six bulk-to-boundary propagators and scales like $\ell^{6 \Delta}$. But the second two have only four bulk-to-boundary propagators, so they scale like $\ell^{4 \Delta}$. They demonstrate how $\phi^3$ can behave like $\phi$ once interactions are taken into account.