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Extremal Couplings in AdS/CFT and Their Effects on Correlation Functions


Core Concepts
This paper investigates the consequences of including "extremal couplings" in the AdS/CFT correspondence, demonstrating that these couplings, while often disregarded, lead to significant effects such as operator mixing and anomalous dimensions in the dual CFT.
Abstract

Bibliographic Information:

Castro, A., & Martinez, P. J. (2024). Revisiting Extremal Couplings in AdS/CFT. [Preprint]. arXiv:2409.15410v2 [hep-th].

Research Objective:

This paper investigates the implications of incorporating "extremal couplings" into the AdS/CFT correspondence, a framework typically neglecting such couplings. The authors aim to understand the consequences of these couplings on the correlation functions within the dual Conformal Field Theory (CFT).

Methodology:

The authors employ a bottom-up approach, focusing on a simplified model of massive scalar fields in Anti-de Sitter (AdS) spacetime with cubic extremal interactions. They utilize an asymptotic cutoff prescription to regulate divergent integrals arising from these couplings and employ holographic renormalization techniques to render the on-shell action finite.

Key Findings:

  • Extremal couplings, where the conformal dimension of one CFT operator matches the sum of the others involved in the interaction, lead to divergences in the AdS vertex.
  • These divergences can be systematically regulated and renormalized using appropriate counterterms.
  • The renormalized theory exhibits non-trivial mixing between single- and double-trace operators in the CFT.
  • The authors calculate the anomalous dimensions of the corrected operators to leading order in perturbation theory, demonstrating the impact of extremal couplings on the CFT spectrum.

Main Conclusions:

The inclusion of extremal couplings in the AdS/CFT correspondence, while leading to initial divergences, can be consistently incorporated through renormalization. This inclusion results in a richer and more intricate structure within the dual CFT, characterized by operator mixing and anomalous dimensions.

Significance:

This work challenges the conventional practice of neglecting extremal couplings in AdS/CFT, highlighting their non-trivial effects on the holographic dictionary. It provides a deeper understanding of the correspondence and opens avenues for exploring new CFT phenomena.

Limitations and Future Research:

The study focuses on a simplified model of scalar fields with cubic interactions. Further research could explore the implications of extremal couplings in more complex scenarios involving higher-spin fields, different interaction terms, and higher-order corrections. Investigating the potential implications for specific top-down AdS/CFT models would also be of interest.

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Quotes
"The AdSd+1/CFTd correspondence is by now a well-established tool to explore perturbative and non-perturbative properties of both conformal field theory (CFTd) and quantum gravity in Anti-de Sitter (AdSd+1)." "Our aim is to show how this divergence is renormalized and the interpretation of the regularized vertex in the CFTd." "The outcome of holographic renormalization is that the generating functional now contains a logarithmic three-point function at a finite distance in configuration space for the dual operators."

Key Insights Distilled From

by Alejandra Ca... at arxiv.org 10-07-2024

https://arxiv.org/pdf/2409.15410.pdf
Revisiting Extremal Couplings in AdS/CFT

Deeper Inquiries

How might the inclusion of extremal couplings affect other aspects of the AdS/CFT correspondence, such as holographic entanglement entropy or the behavior of Wilson loops?

The inclusion of extremal couplings in the AdS/CFT correspondence could potentially lead to intriguing modifications in various holographic observables, including entanglement entropy and Wilson loops. Here's a breakdown of how these effects might manifest: Holographic Entanglement Entropy: Anomalous contributions to entanglement entropy: Extremal couplings, by inducing anomalous dimensions in the CFT, could introduce logarithmic corrections to the entanglement entropy. This stems from the fact that entanglement entropy is sensitive to the scaling dimensions of operators, and any modification in these dimensions due to extremal couplings would naturally percolate to the entanglement structure. Modified entanglement wedge reconstruction: The entanglement wedge reconstruction proposal posits a correspondence between subregions in the bulk and their entangled counterparts on the boundary. Extremal couplings, by altering the bulk dynamics, might lead to subtle changes in the entanglement wedge, potentially affecting the reconstruction of bulk operators from boundary data. Wilson Loops: Shifted quark-antiquark potential: In holographic setups, the expectation value of a Wilson loop often computes the potential energy between a quark-antiquark pair. Extremal couplings, through their impact on the bulk geometry or interactions, could modify this potential, potentially leading to different screening or confinement behaviors. Modified cusp anomalous dimension: The cusp anomalous dimension, characterizing the divergence of a Wilson loop with a cusp, is another quantity sensitive to the field content and interactions in the CFT. Extremal couplings could introduce corrections to this dimension, signaling modifications in the UV behavior of the theory. Exploring these effects: Investigating these potential consequences would require explicit calculations within specific AdS/CFT models incorporating extremal couplings. Techniques like the Ryu-Takayanagi formula for entanglement entropy and the holographic dictionary for Wilson loops would be instrumental in quantifying these modifications.

Could there be alternative interpretations of the logarithmic divergences arising from extremal couplings, potentially leading to different physical consequences in the CFT?

While the paper interprets the logarithmic divergences arising from extremal couplings as a signal of operator mixing and anomalous dimensions, alternative interpretations might exist, potentially leading to different physical consequences. Here are a few speculative possibilities: Emergent symmetries: The logarithmic divergences could hint at the emergence of new symmetries in the CFT at the extremal point. These symmetries might not be manifest in the original Lagrangian description but could arise dynamically due to the specific structure of extremal couplings. Non-local effects: The logarithmic behavior might be a manifestation of non-local effects in the CFT. Extremal couplings could introduce subtle long-range correlations that are not captured by the usual local operator product expansion, leading to logarithmic corrections in correlation functions. Flow to a different CFT: It's conceivable that the extremal coupling triggers a renormalization group flow to a different conformal fixed point. The logarithmic divergences could then be interpreted as signatures of this flow, with the anomalous dimensions characterizing the approach to the new fixed point. Further investigation: To explore these alternative interpretations, one could: Analyze higher-order corrections: Studying higher-order corrections in the extremal coupling might reveal additional structures or patterns in the logarithmic divergences, providing further clues about their origin. Explore different regularization schemes: Employing different regularization and renormalization schemes could shed light on the universality of the logarithmic behavior and potentially uncover scheme-dependent artifacts. Search for dualities: Investigating potential dualities of the CFT with extremal couplings might provide new perspectives on the logarithmic divergences and their physical implications.

What are the implications of this work for understanding the emergence of spacetime and gravity from a non-gravitational, conformal field theory?

This work, by exploring the unconventional realm of extremal couplings in AdS/CFT, offers intriguing hints about the emergence of spacetime and gravity from a non-gravitational CFT. Here's a perspective on the potential implications: Beyond conventional AdS/CFT: Traditional AdS/CFT examples typically avoid extremal couplings, focusing on setups where the bulk interactions are well-behaved. This work suggests that venturing beyond these conventional examples might unveil richer and potentially more realistic holographic dualities. New mechanisms for gravity: The emergence of logarithmic divergences and anomalous dimensions from extremal couplings suggests novel mechanisms for generating effective gravitational interactions in the bulk. These mechanisms could be relevant for understanding how gravity arises from a non-gravitational CFT in more general holographic scenarios. Quantum corrections to geometry: The need for counterterms to regulate the extremal couplings highlights the importance of quantum corrections in shaping the bulk geometry. These corrections, captured by the anomalous dimensions in the CFT, suggest a deep interplay between quantum entanglement and the emergence of spacetime. Future directions: To further explore these implications, one could investigate: Extremal couplings in different dimensions: Studying extremal couplings in different AdS/CFT models, particularly in lower dimensions where quantum gravity effects are more pronounced, could provide valuable insights into the emergence of spacetime. Connection to other approaches: Exploring connections between extremal couplings and other approaches to quantum gravity, such as tensor networks or group field theory, could lead to a more unified understanding of the holographic emergence of spacetime. Phenomenological implications: While speculative, it's worth exploring whether extremal couplings and their associated logarithmic corrections could have any phenomenological implications for cosmology or condensed matter systems exhibiting holographic duality.
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