New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators (1201.6449v2)

Abstract: We consider correlation functions of the stress-tensor or a conserved current in AdS_{d+1}/CFT_d computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the BCFW-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including d=3. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the (d+1)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the d-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon or graviton amplitude appears as its coefficient.

• The paper introduces new recursion relations for efficiently computing AdS/CFT correlators across dimensions, including the critical d=3 case.
• It outlines a method to extract flat-space S-matrix elements from boundary correlators, maintaining validity even through loop-level corrections.
• The study enhances computational techniques in quantum gravity and holography, paving the way for broader applications in theoretical physics.

New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators

The paper "New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators" by Suvrat Raju introduces new methodologies for computing correlation functions of the stress tensor and conserved currents within AdS/CFT frameworks. It specifically offers advancements over previously established BCFW-type recursion relations, extending their applicability to all dimensions, including the critically important case of $d=3$. Simultaneously, the author proposes a novel technique to discern flat-space S-matrix elements from AdS/CFT correlators, thus bridging the gap between boundary correlators and flat-space amplitudes, even incorporating loop-level computations.

Overview

AdS/CFT correspondence provides a theoretical basis for computing boundary CFT correlators starting from perturbative bulk quantum field theories. Despite its conceptual clarity, practical applications are often hampered by the complexity of gravitational interactions. This complexity arises primarily because the Hilbert action of gravity in the bulk results in infinitely complex interaction vertices, complicating standard calculations.

The paper tackles these difficulties head-on by introducing new recursion relations that mitigate these computational burdens, allowing for more efficient calculations of AdS/CFT correlators at tree level. It underscores two main advancements:

1. Recursion Relations: The paper develops new recursion relations analogous to those used for gluon and graviton amplitudes in flat space (Risager's recursion), but tailored for AdS/CFT. These are valid across all dimensions and are particularly advantageous for calculations when $d=3$.
2. S-Matrix Extraction Methodology: It provides a method for deriving the flat-space S-matrix elements from momentum-space AdS/CFT correlators. At its core, this method shows that flat-space amplitudes can be determined from specific singularity coefficients of boundary correlators, holding true even with higher-order loop corrections.

Numerical Results and Claims

The recursion relations presented are claimed to reliably compute correlators consistent with expected singularity patterns, with flat-space gluon or graviton amplitudes manifesting as their coefficients. These results are claimed to be valid through higher-order loop calculations, a significant claim given the traditionally increased difficulty of loop computations.

Implications and Future Work

The techniques introduced promise substantial improvements in computational efficiency and breadth of applicability for determining correlators in AdS/CFT. This could lead to deeper insights into the AdS/CFT correspondence itself, facilitating explorations into quantum gravity and high-energy physics. The ability to calculate loop-level amplitudes and directly connect boundary correlators to flat-space scattering processes offers potentially transformative implications for theoretical and computational physics.

While the paper remains grounded in its contributions to the current understanding of AdS/CFT, its methodologies could inspire future research directions, potentially even in holographically dual theories with more complex interactions. The approach could be pivotal in extending the range of applications for holographic theories beyond traditional settings, possibly even including higher derivative corrections or unconventional dualities.

Conclusion

This paper advances the methodology for calculating correlators in AdS/CFT, enhancing the computation of gravitational and gluon interactions in arbitrary dimensions. By elucidating a clear path from boundary correlators to flat-space scattering amplitudes, Raju has paved the way for further exploration and application within the field of quantum field theory and string theory. These foundational improvements stand to significantly impact ongoing and future investigations into the holographic principles that underpin modern theoretical physics.