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.
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