Holographic four-point functions in the (2, 0) theory

Journal of High Energy Physics, Jun 2018

Abstract We revisit the calculation of holographic correlators for eleven-dimensional supergravity on AdS7 × S4. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity effective action. By an extension of the position space approach developed in [1, 2] for the AdS5 × S5 background, we compute four-point correlators of one-half BPS operators for identical weights k = 2, 3, 4. The k = 2 case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function.

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Holographic four-point functions in the (2, 0) theory

Accepted: June Holographic four-point functions in the (2; 0) theory Leonardo Rastelli 0 1 Xinan Zhou 0 1 Theory, Scattering Amplitudes 0 Stony Brook , 11794, NY , U.S.A 1 C.N. Yang Institute for Theoretical Physics, Stony Brook University We revisit the calculation of holographic correlators for eleven-dimensional S4. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity e ective action. By an extension of the position space approach developed in [1, 2] for the AdS5 S5 background, we compute four-point correlators of one-half BPS operators for identical weights k = 2; 3; 4. The k = 2 case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function. AdS-CFT Correspondence; Conformal and W Symmetry; Conformal Field - supergravity on AdS7 1 Introduction 2 3 4 Mellin space 3.1 3.2 3.3 3.4 4.1 4.2 4.3 Superconformal kinematics of four-point functions Position space The position space method The k = 2 four-point function The k = 3 four-point function The k = 4 four-point function Consistency conditions and the algebraic problem Solutions from the position space method A R-symmetry polynomials B Fixing the overall constant C Four-point functions of the Wn!1 algebra servables to be computed using the AdS/CFT dictionary, but only recently [1, 2] e cient calculational methods have begun to be developed. The traditional recipe is based on a perturbative expansion in Witten diagrams, which becomes very cumbersome (already at tree level) for n-point correlators with n > 4. Prior to our work, only a few four-point correlators of Kaluza-Klein (KK) modes were known, to wit (focussing for de niteness on the maximally supersymmetric backgrounds): a handful of cases in the AdS5 S5 background [3{9]; just the four-point function of the lowest KK mode (the stress-tensor multiplet) in the AdS7 S4 background [10]; and no results whatsoever in the AdS4 S7 background. The traditional method has two sources of computational complexity: the need for explicit expressions of the vertices in the supergravity e ective action; and the proliferation of exchange Witten diagrams as the KK level is increased. In [1, 2] we introduced new calculational tools to circumvent these di culties, for the case of AdS5 A rst approach, which we refer to as the \position space method", leverages the special feature of the AdS5 S5 background that all exchange Witten diagrams can be written as nite sums of contact diagrams. One can then write an ansatz for the four-point correlator S5 supergravity. { 1 { as a sum of contact diagrams, and determine their relative coe cients by imposing superconformal symmetry, with no need for a detailed knowledge of the e ective supergravity action. While simpler than the standard perturbative recipe, the position space method also quickly runs out of steam as the KK level is increased. What's worse, the answer takes a completely unintuitive form, with no simpe general pattern. The second, more powerful approach of [1, 2] uses the Mellin representation of conformal correlators [11, 12]. Tree-level holographic correlators in AdS5 S5 are rational functions of Mandelstam-like invariants, with poles and residues controlled by OPE factorization, in close analogy with tree-level at space scattering amplitudes. Superconformal symmetry is made manifest by solving the superconformal Ward identity in terms of an \auxiliary" Mellin amplitude. The consistency conditions that this amplitude must satisfy de ne a very constrained algebraic problem, which very plausibly admits a unique solution. While the position space method is implemented on a case-by-case basis for di erent correlators, the Mellin algebraic problem takes a universal form. We were able to solve the problem in one fell swoop for all half-BPS four-point function with arbitrary weights | a feat extremely di cult to replicate in position space. The goal of this note is to extend these techniques to eleven-dimensional supergravity on AdS7 S4. This is a background of extraordinary physical interest, as it provides a dual description of the mysterious six-dimensional (2; 0) theory at large n, and a prime target for our methods, since we expect maximal supersymmetry to constrain the treelevel holographic correlators uniquely. At a more technical level, this background enjoys the same \truncation conditions" as AdS5 S5, such that exchange Witten diagrams can be written as nite sums of contact diagrams | equivalently, tree-level Mellin amplitudes are rational functions of the Mandelstam invariants. By contrast, the truncation conditions do not hold for the AdS4 S7 background, and new too (...truncated)


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Leonardo Rastelli, Xinan Zhou. Holographic four-point functions in the (2, 0) theory, Journal of High Energy Physics, 2018, pp. 87, Volume 2018, Issue 6, DOI: 10.1007/JHEP06(2018)087