Looking for a bulk point

Abstract: We consider Lorentzian correlators of local operators. In perturbation theory, singularities occur when we can draw a position-space Landau diagram with null lines. In theories with gravity duals, we can also draw Landau diagrams in the bulk. We argue that certain singularities can arise only from bulk diagrams, not from boundary diagrams. As has been previously observed, these singularities are a clear diagnostic of bulk locality. We analyze some properties of these perturbative singularities and discuss their relation to the OPE and the dimensions of double-trace operators. In the exact nonperturbative theory, we expect no singularity at these locations. We prove this statement in 1+1 dimensions by CFT methods.

• The paper distinguishes between perturbative and non-perturbative singularities in CFTs, proving their absence in non-perturbative 1+1 configurations.
• It derives precise position-space Landau rules and examines the effects of D-instantons and finite α corrections on bulk singularity resolution.
• It extends the analysis across dimensions, showing that bulk singularities depend on dimensionality and are absent in 1+1 boundary theories, reinforcing the holographic principle.

Analyzing Bulk-Point Singularities in Lorentzian Correlators: A Detailed Examination

The paper authored by Juan Maldacena, David Simmons-Duffin, and Alexander Zhiboedov explores the intriguing topic of singularities in Lorentzian correlators, specifically distinguishing between those arising in weakly coupled theories versus those present in theories with gravity duals. The focus of the paper is to investigate and characterize “bulk-point singularities,” a concept significant in understanding the locality of interactions in conformal field theories (CFTs) with gravity duals.

Key Insights and Contributions

The crux of the paper revolves around the emergence of singularities in Lorentzian correlators, drawing a line between the contributions from boundary versus bulk diagrams. In traditional quantum field theories (QFTs), singularities are rooted in the ability to construct Landau diagrams using null lines, which are indicative of specific interaction points in spacetime where conservation rules hold. The paper argues that in theories with AdS/CFT duality, some singularities originate solely from bulk Landau diagrams and remain unmatched by any boundary counterparts. This distinction serves as a probe for assessing bulk locality.

1. Perturbative vs. Non-Perturbative Singularities: The analysis focuses on how singularities manifest differently in perturbative expansions of QFTs compared to results from non-perturbative analyses. In particular, the authors prove the absence of singularities in exact non-perturbative configurations within 1+1 dimensions using CFT techniques.
2. Analytical Tools and Geometric Considerations: The paper provides a meticulous derivation of position-space Landau rules analogous to momentum-space analogs. D-instantons and stringy corrections are assessed for their impacts on bulk singularities, with discussions indicating that finite α corrections could avert these singularities, as corroborated by examples considered within the framework.
3. Extended Dimensional Analysis: The research extends the discussion to different dimensional frameworks, comparing the presence and absence of bulk singularities across 1+1, 2+1, and higher dimensions. Importantly, it is demonstrated that while some singularities can emerge only in bulk dimensions, such as in a 2+1 framework, they are entirely absent in 1+1 boundary theories, emphasizing the dimensional dependence of these features.
4. Singular Localization on Cross-Ratio Space: Emphasis is placed on the significance of cross-ratio spaces where singularities manifest in a perturbative manner, and the relevance of bulk configurations to such mathematical analysis. The authors define hypersurfaces within those spaces, identifying conditions under which singularities are present or eliminated in perturbative treatments.
5. Explorations into Instanton Effects and Regge Physics: Addressing instanton corrections, the authors explore how these contributions manifest similarly at both weak and strong coupling. They theorize potential resolutions to singularities using high-energy scattering insights in AdS space, relying on analytic continuation techniques pertinent to Regge physics.

Implications and Future Directions

This paper's exploration of bulk-point singularities is pivotal in advancing our understanding of conformal field theory dynamics, especially regarding how these theories mirror gravitational interactions under the holographic principle. The paper's mathematical rigor in analyzing correlator singularities lays groundwork for future studies in CFT with applications in understanding the emergent gravity, the spacetime continuum, and the fundamental structure of quantum gravity.

Theoretical implications suggest that further computational insights into the planar correlators at strong coupling could leverage integrability techniques, potentially allowing for the direct observation of emergent singularities. The implications for experimental physics could harness these insights to delineate quantum-to-classical transitions within the context of gravity. The paper encourages exploration into suppressed singularities at finite gravitational couplings, inviting analyses reflective of classical-black-hole phenomena.

Conclusion

Maldacena, Simmons-Duffin, and Zhiboedov offer a comprehensive study that ties mathematical rigorousness with theoretical physics, advancing our comprehension of bulk interactions via singular structures. By differentiating between bulk and boundary effects, this work continues to bridge the gap between quantum field theory and string-theoretic approaches to gravity, reinforcing a more unified understanding of the governing principles in higher-dimensional physics. As this research unfolds, persistent interrogation into higher order corrections and deeper structural singularities promises to enrich the broader discourse in theoretical physics.