Holographic Correlation Functions from Excised AdS Geometries

Core Concepts

This paper proposes a novel holographic method for computing correlation functions of heavy operators in conformal field theories (CFTs) by calculating the on-shell action of excised AdS geometries, specifically focusing on AdS3 solutions and comparing the results with those obtained from backreacted geometries.

Abstract

Lai, T., Sun, Y., & Tian, J. (2024). Holographic correlation functions from wedge. arXiv preprint arXiv:2411.12420.

This paper aims to develop a new holographic method for calculating correlation functions of heavy operators in conformal field theories (CFTs) using the on-shell action of excised AdS geometries.

Key Insights Distilled From

Holographic correlation functions from wedge

by Tengzhou Lai... at arxiv.org 11-20-2024

https://arxiv.org/pdf/2411.12420.pdf

Deeper Inquiries

How does this method of calculating correlation functions from excised geometries extend to higher-point functions and more complex CFT operators?

Extending the excised geometry method to higher-point functions and more complex CFT operators presents exciting challenges and potential rewards:
Higher-Point Functions:

Multiple Wedges: For an n-point function, one would intuitively imagine excising n wedges from the AdS spacetime, each associated with a heavy operator insertion. The challenge lies in precisely defining the geometry of these multiple wedges, ensuring their intersections and boundaries are consistent with the desired CFT correlator.
New Corner Contributions: The intersection of multiple wedges introduces new corner contributions to the on-shell action. A careful analysis of these contributions, potentially requiring generalizations of the Hayward term, would be crucial for obtaining the correct CFT result.
Computational Complexity: Calculating the on-shell action for intricate multi-wedge geometries will likely be computationally demanding, potentially requiring numerical methods or novel analytical techniques.
Complex CFT Operators:

Beyond Conical Defects: While conical defects effectively capture the backreaction of heavy scalar operators, more complex operators (e.g., operators with spin or those in non-trivial representations of the CFT symmetry group) might require different geometric constructs.
Boundary Conditions: The choice of boundary conditions on the EOW branes would need to reflect the specific properties of the dual CFT operators. This might involve incorporating additional boundary degrees of freedom or modifying the brane actions.
Potential Strategies:

Perturbative Approach: Starting with a known lower-point function, one could perturbatively introduce additional wedges, systematically calculating the corrections to the on-shell action.
Exploiting Symmetries: Symmetries of the CFT correlator could simplify the geometric construction. For example, conformal invariance might restrict the allowed configurations of multiple wedges.
Numerical Techniques: Numerical relativity tools could prove invaluable for constructing and analyzing complex excised geometries, especially in cases where analytical solutions are intractable.

Could the discrepancy in on-shell actions between different coordinate systems be attributed to a specific choice of boundary counterterms or a more fundamental aspect of the AdS/CFT dictionary?

The discrepancy in on-shell actions between different coordinate systems, such as those observed between the transformed conical coordinates and Fefferman-Graham (FG) coordinates, raises a fundamental question about the AdS/CFT dictionary:
Boundary Counterterms and Ambiguities:

Role of Counterterms: In AdS/CFT, boundary counterterms are crucial for rendering the on-shell action finite and well-defined. Different coordinate systems might necessitate different choices of counterterms to properly account for divergences near the AdS boundary.
Coordinate Dependence of Counterterms: The form of boundary counterterms can explicitly depend on the choice of boundary metric. Since different coordinate systems induce different boundary metrics, the corresponding counterterms might not be equivalent, leading to discrepancies in the on-shell action.
More Fundamental Aspects:

Choice of Boundary Conditions: The discrepancy might hint at a subtle dependence of the AdS/CFT dictionary on the choice of boundary conditions. Different coordinate systems might implicitly impose different boundary conditions on the bulk fields, affecting the on-shell action.
Quantum Corrections: It's conceivable that quantum corrections to the AdS/CFT dictionary play a role. The classical on-shell action might receive non-trivial contributions from quantum fluctuations, and these contributions could be sensitive to the choice of coordinates.
Resolving the Discrepancy:

Systematic Analysis of Counterterms: A thorough investigation of the boundary counterterms required in different coordinate systems is essential. This analysis should carefully account for the coordinate dependence of the boundary metric and the induced counterterms.
Comparison with CFT Results: Ultimately, the validity of different coordinate systems should be judged by their ability to reproduce known CFT results. If a discrepancy persists after accounting for counterterms, it might indicate a need to refine our understanding of the AdS/CFT dictionary.

How can this approach be applied to study other holographic observables, such as entanglement entropy or Wilson loops, and what new insights might it offer?

The excised geometry approach holds promise for studying other holographic observables beyond correlation functions, potentially offering fresh perspectives on entanglement entropy and Wilson loops:
Entanglement Entropy:

Ryu-Takayanagi and Beyond: The Ryu-Takayanagi (RT) formula relates entanglement entropy in the CFT to the area of minimal surfaces in the bulk. Excising a wedge from AdS modifies the geometry and thus alters the minimal surfaces relevant for entanglement entropy calculations.
Defect Entanglement: Excised geometries naturally introduce defects into the AdS spacetime. This approach could provide insights into the entanglement entropy associated with these defects, a topic of significant interest in condensed matter and quantum information theory.
Quantum Corrections: Studying how entanglement entropy changes under wedge excision might offer clues about quantum corrections to the RT formula, particularly in the presence of heavy operators or defects.
Wilson Loops:

Geometric Interpretation: Wilson loops in the CFT are dual to the area of minimal surfaces in AdS that are bounded by the loop on the boundary. Excising a wedge would modify these minimal surfaces, potentially leading to new insights into the behavior of Wilson loops in strongly coupled gauge theories.
Confinement and Screening: The way in which Wilson loop areas change under wedge excision might provide information about confinement and screening mechanisms in the dual gauge theory. For example, the presence of a wedge could mimic the effects of a quark-antiquark potential.
Non-Local Observables: Wilson loops are inherently non-local observables, and their behavior can be sensitive to the global structure of spacetime. Excised geometries offer a controlled way to probe the interplay between non-locality and the presence of heavy operators or defects.
New Insights and Challenges:

Universal Features: This approach might reveal universal features of entanglement entropy and Wilson loops in the presence of heavy operators or defects, independent of the specific CFT.
Computational Tools: Developing efficient computational tools for calculating minimal surfaces in excised geometries will be crucial for extracting quantitative predictions.
Conceptual Understanding: A deeper conceptual understanding of how the excised geometry approach relates to other holographic techniques for calculating entanglement entropy and Wilson loops is essential for fully exploiting its potential.