On superconformal four-point Mellin amplitudes in dimension d > 2

Abstract

We present a universal treatment for imposing superconformal constraints on Mellin amplitudes for SCFT d with 3 ≤ d ≤ 6. This leads to a new technique to compute holographic correlators, which is similar but complementary to the ones introduced in [1, 2]. We apply this technique to theories in various spacetime dimensions. In addition to reproducing known results, we obtain a simple expression for next-next-to-extremal fourpoint functions in AdS 7 × S 4 . We also use this machinery on AdS 4 × S 7 and compute the first holographic one-half BPS four-point function. We extract the anomalous dimension of the R-symmetry singlet double-trace operator with the lowest conformal dimension and find agreement with the 3d N = 8 numerical bootstrap bound at large central charge.

AdS4/CFT3 for unprotected operators

“…0,0,0 was already obtained by [16] using the superconformal primary (2, 0) [0000] . We now move on to the second lowest twist operators (A, 0) j,1,q , which has ∆ (0) (A,0,m) j,1,q = j + 4, i.e.…”

“…Progress was made recently in [16], which computed the tree level supergravity contribution to the Mellin space [17] holographic four point function of the lowest scalar in the stress tensor multiplet. 3 This correlator was then expanded in terms of conformal blocks to read off the 1/c T correction to the scaling dimension of the lowest unprotected operator that appears, which was matched to the large c T limit of ABJ(M) theory 4 computed from the N = 8 numerical bootstrap [37].…”

“…In section 3, we review the computation of the leading order CFT data in this correlator. In section 4, we present an algorithm for extracting CFT data from maximally supersymmetric AdS 4 Mellin amplitudes, and apply it to the tree level amplitude computed in [16]. We then compare previously computed analytical and numerical results.…”

“…The tree level O Stress four-point function receives contributions from contact and exchange Witten diagrams of single trace operators. This correlator was computed explicitly in Mellin space in [16]. We will now review this Mellin space amplitude, and then extract all the relevant CFT data from it.…”

“…The poles of the Gamma functions precisely capture the contribution of double trace operators in the OPE. The contributions from the single trace exchange and the contact diagrams were computed by [16] using the superconformal Ward identities [55] and the assumption that the contact diagram is linear in s, t, u, which is implied by the two derivative Einstein-Hilbert term in the supergravity action. The resulting Mellin amplitude takes the form…”