Abstract
The anti-de Sitter/conformal field theory (AdS/CFT) correspondence is a remarkable tool for analytically studying strongly coupled physics. Thanks to the AdS/CFT correspondence, strongly correlated quantum systems can be understood as weakly coupled gravity theories, which live in a holographic spacetime with an extra emergent dimension and constant negative curvature. Fundamental observables such as correlation functions are identified with scattering amplitudes in curved spacetime, which, in principle, can be computed by using standard perturbation theory. Unfortunately, such holographic calculations are notoriously difficult, even for tree-level processes involving four external particles. Despite relentless efforts over the past two decades, a full solution to this problem was not found. In this article, we introduce a powerful new method that solves this long-standing problem. We give a closed-form formula for all such four-point functions in a class of theories that constitute the best-known paradigms of AdS/CFT. These models exhaust the theories compatible with maximal supersymmetry, and they live in three, four, and six dimensions. Pivotal to our construction is the use of symmetries. We show that in a judiciously chosen limit, symmetry principles dictate a drastic simplification in holographic correlators, allowing them to be directly computed. Having solved this limit, we further show that the full correlators can be recovered from this special configuration by using only symmetries. In addition to providing valuable explicit expressions that have a wide range of applications in AdS/CFT, our analysis leads to several important conceptual lessons. Our results point out remarkable simplicities underlying the holographic correlators, as well as concrete ways to search for such structures. Moreover, our construction identifies previously unknown elegant underlying organizing principles for holographic correlators. These qualitative features of holographic correlators also echo the exciting progress in the scattering amplitude program in flat space, suggesting tantalizing prospects of future cross-fertilization of ideas.
- Received 14 September 2020
- Accepted 25 January 2021
DOI:https://doi.org/10.1103/PhysRevX.11.011056
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The anti–de Sitter/conformal field theory (AdS/CFT) correspondence is a powerful tool that connects theories of quantum gravity with quantum field theories, which describe elementary particles and forces. The conjectured relationship between the two types of theories allows researchers to translate complex problems in one theory into a more tractable problem in the language of the other. However, despite decades of effort, some of the necessary computations are still often hopelessly complicated. Here, we present a new mathematical framework that greatly simplifies these computations.
We can think of AdS space as a box, and the CFT “lives” on its surface. The AdS/CFT duality tells us that we can describe what happens inside the box by looking at the surface and vice versa. In particular, scattering amplitudes inside the box are mapped to certain mathematical objects (correlators) on the surface of the box.
Scattering amplitudes describe the possible outcomes when two particles collide. They are hard to compute, but they are important because they encode the way particles interact in nature, and they also can be directly measured at particle colliders. Here, we elucidate the elegant underlying organizing principles of the correlators and write down all such amplitudes in the class of best-known paradigms of AdS/CFT.
The results provide a wealth of theoretical data that are essential, for example, for analyzing quantum features of gravity in AdS. More importantly, our strategy provides a valuable lesson in further harnessing the computational power of AdS/CFT: Effectively using symmetry principles strips away the intimidating superficial complexities and allows us to focus on the core of the problem, which is often surprisingly simple.
