A CFT Perspective on Gravitational Dressing and Bulk Locality (1608.08977v2)

Abstract: We revisit the construction of local bulk operators in AdS/CFT with special focus on gravitational dressing and its consequences for bulk locality. Specializing to 2+1-dimensions, we investigate these issues via the proposed identification between bulk operators and cross-cap boundary states. We obtain explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, and find that they are free of non-local branch cuts but do have non-local poles. We recover the HKLL recipe for restoring bulk locality for interacting fields as the outcome of a natural CFT crossing condition. We show that, in a suitable gauge, the cross-cap states solve the bulk wave equation for general background geometries, and satisfy a conformal Ward identity analogous to a soft graviton theorem, Virasoro symmetry, the large N conformal bootstrap and the uniformization theorem all play a key role in our derivations.

• The paper demonstrates that gravitational dressing can be understood via cross-cap boundary states, yielding bulk correlators with non-local poles but free from branch cuts.
• It recovers the HKLL reconstruction method from CFT crossing conditions, providing a theoretically sound approach to secure local bulk operator construction.
• The study highlights the role of Virasoro symmetry and conformal Ward identities, drawing analogies with soft graviton theorems to advance holographic insights.

A CFT Perspective on Gravitational Dressing and Bulk Locality

The paper provides a comprehensive exploration of the construction of local bulk operators within the framework of AdS/CFT correspondence, particularly focusing on aspects of gravitational dressing and its impact on bulk locality. Specializing in the context of AdS${}_3$/CFT${}_2$, the authors investigate how these concepts can be understood through the identification between bulk operators and cross-cap boundary states. The exploration is centered around the derivation of explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, leading to crucial insights into the nature of non-local poles versus branch cuts in these functions.

Key Findings and Claims:

1. Correlation Functions Without Non-local Branch Cuts: The paper explores correlation functions of bulk fields interacting with boundary stress tensor insertions and finds that while these functions lack non-local branch cuts—a typical concern in gravitational holography—they do exhibit non-local poles. This distinction is essential for understanding the non-local characteristics inherent in quantum gravitational theories.
2. Recovery of the HKLL Recipe: The paper revisits the HKLL (Hamilton-Kabat-Lifschytz-Lowe) approach to reconstructing bulk locality from CFT data in the context of interacting fields. It identifies how the HKLL prescription can naturally arise from CFT crossing conditions, offering a theoretically justified procedure to ensure bulk locality.
3. Role of Cross-cap States in Bulk Wave Equations: The authors demonstrate that, within an appropriate gauge, cross-cap states solve the bulk wave equation across general background geometries. This emphasizes the adaptability of cross-cap states to various configurations, upholding bulk dynamics, and addressing different boundary conditions.
4. Uniformization Theorem and Virasoro Symmetry: A key insight is the role of Virasoro symmetry, the large $N$ conformal bootstrap, and the uniformization theorem in these derivations. These components are pivotal in managing non-localities in the reconstructed bulk operators and ensuring they conform to expected physical constraints.
5. Conformal Ward Identity as a Soft Graviton Theorem Analogue: The paper highlights that the conformal Ward identities satisfied by cross-cap states are akin to soft graviton theorems in other gravitational settings, reflecting essential symmetry properties of gravitational amplitudes and their conservation laws.

Implications and Future Directions:

The findings have significant implications for advancing our understanding of quantum gravity and the holographic principle. The insights into how dressed bulk operators, viewed through the lens of CFT, adapt to different background geometries could pave the way to resolving longstanding puzzles regarding bulk locality and symmetry in gravitational settings.

Moreover, the paper invites further exploration into higher-dimensional extensions, where similar dualities might have rich structures due to enhanced group symmetries and additional degrees of freedom. The approach could also deepen the understanding of entanglement structures in holographic theories and their fields of influence.

In exploring future developments, the results here could stimulate new research examining alignment or deviations when applying analogous constructs in other holographic frameworks or non-AdS settings. Furthermore, the potential applicability to quantum error correction codes within holography underscores a deeper unifying principle that could provide a robust framework for understanding quantum information dynamics in gravity.

Overall, the treatment of gravitational dressing through cross-cap states in CFT offers a nuanced and mathematically consistent framework, leading to consequential descriptions of how bulk theories may manifest while constrained by boundary conformal theories, marking a significant stride in bridging the gap between abstract theoretical constructs and tangible physical phenomena.