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Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT (1104.2621v2)

Published 13 Apr 2011 in hep-th

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.

Citations (188)

Summary

  • The paper demonstrates that operator dictionaries are equivalent in AdS/CFT but fail to match in dS/CFT due to differing boundary conditions.
  • It employs analytic continuation to relate Euclidean AdS wave functions to those in dS, uniting distinct geometric scenarios.
  • The study extends its framework to perturbative dynamical gravity, offering insights into holographic renormalization and quantum gravity.

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

The paper "Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT" by Daniel Harlow and Douglas Stanford explores the formulations and implications of different operator dictionaries within the context of the AdS/CFT and dS/CFT correspondences, significant frameworks in theoretical physics aimed at understanding quantum gravity through holographic principles.

The AdS/CFT correspondence provides a duality between a theory of gravity in anti-de Sitter (AdS) space and a conformal field theory (CFT) on its boundary. Central to its application is the construction of dictionaries that translate physical quantities in one space to the dual space. The dual AdS/CFT correlators can be derived via two principal methods: differentiating the bulk partition function with respect to boundary conditions (GKPW dictionary) or extrapolating bulk correlation functions to the boundary (BDHM dictionary). In this paper, the authors revisit these methods to assess their equivalence, particularly in interacting theories, by scrutinizing the renormalization of bulk composite operators within the path integrals of the bulk.

Interestingly, the paper concludes that while these dictionaries are equivalent in the context of AdS/CFT, this equivalence does not extend to dS/CFT, a proposed duality involving de Sitter (dS) space. This distinction underscores a critical difference in the treatment of boundary conditions between AdS, which embodies a time-like boundary allowing causal constraint, versus dS, where the space-like boundaries inherently allow for more complex fluctuation dynamics, leading to non-equivalence in the dictionary formulations.

The authors demonstrate that for Euclidean AdS, the wave function analytically continues to that for dS with Euclidean initial conditions. This intriguing result provides a unified perspective that connects these two geometries through analytic continuation, although they serve different physical scenarios characterized by opposite-sign cosmological constants.

The paper then extends the discussion to include perturbative dynamical gravity, suggesting how the results obtained in fixed backgrounds could be generalized while maintaining the core principles of holographic renormalization. This again emphasizes the flexibility and potential of these correspondence frameworks in broader and more complex setups.

From a numerical standpoint, the authors provide a robust framework for interpreting these dictionary formulations, reinforced by an analytical approach that offers insight into the subtle mechanics of field behavior near boundaries in these spacetimes. It's noteworthy that the implication of their findings goes beyond immediate computational results, offering a pathway for more refined theoretical developments in the understanding of inequivalent dictionary formulations at cosmological boundaries.

In essence, Harlow and Stanford's work is pivotal in distinguishing how boundary dynamics in AdS and dS lead to fundamentally different physical interpretations in holographic dualities. There remain broad implications for quantum gravity theories, especially in regards to how they can be formulated to align more closely with the observed universe, contrasting with the traditional AdS setups. The results encourage further exploration into alternative holographic constructs, potentially offering new insights into the early expansion of the universe, inflationary models, and the lasting mysteries surrounding dark energy and the cosmological constant problem. Future research may provide deeper elucidation on the nature of these dualities, not only enhancing our theoretical understanding but also potentially informing observational cosmology.