We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on AdS 5 × S 5 . Our approach relies entirely on general consistency conditions and maximal supersymmetry. We discuss two related methods, one in position space and the other in Mellin space. The position space method is based on the observation that the holographic four-point correlators of onehalf BPS single-trace operators can be written as finite sums of contact Witten diagrams. We demonstrate in several examples that imposing the superconformal Ward identity is sufficient to fix the parameters of this ansatz uniquely, avoiding the need for a detailed knowledge of the supergravity effective action. The Mellin space approach is an "onshell method" inspired by the close analogy between holographic correlators and flat space scattering amplitudes. We conjecture a compact formula for the four-point correlators of one-half BPS single-trace operators of arbitrary weights. Our general formula has the expected analytic structure, obeys the superconformal Ward identity, satisfies the appropriate asymptotic conditions and reproduces all the previously calculated cases. We believe that these conditions determine it uniquely.

We revisit the calculation of holographic correlation functions in IIB supergravity on AdS5 × S 5 . Results for four-point functions simplify drastically when expressed in Mellin space. We conjecture a compact formula for the four-point functions of one-half BPS single-trace operators of arbitrary weight. Our methods rely on general consistency conditions and eschew detailed knowledge of the supergravity effective action.PACS numbers: 11.25. Tq, 11.15.Pg, 11.55.Bq Introduction. Despite almost two decades of relentless efforts, we are still far from harnessing the full computational power of the AdS/CFT correspondence. In the canonical duality [1][2][3] between N = 4 super Yang-Mills (SYM) theory and IIB string theory on AdS 5 × S 5 , the bulk description is most tractable in the classical supergravity regime, which describes planar SYM theory at large 't Hooft coupling. Supergravity is however still a complicated non-linear theory, and only the simplest observables have been computed so far. In this letter we revisit the holographic calculation of four-point correlation functions of one-half BPS single-trace operators [4]. In the supergravity limit, there is a straightforward algorithm that computes them as a sum of tree-level Witten diagrams, whose vertices are encoded in the AdS 5 effective action [5] obtained by Kaluza-Klein (KK) reduction of IIB supergravity on S 5 .

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.

A natural question about Quantum Field Theory is whether there is a deformation to a trivial gapped phase. If the underlying theory has an anomaly, then symmetric deformations can never lead to a trivial phase. We discuss such discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We emphasize the role of charge conjugation symmetry in these anomalies; for example, we obtain nontrivial constraints on the degrees of freedom that live on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions. In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the Abelian Higgs model is dual to the Ising model. We also study variations of the Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical particle of unit charge. These models have a center symmetry and additional discrete anomalies. In the absence of a dynamical unit charge particle, the Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These models without a unit charge particle exhibit a remarkably persistent order: we prove that the system cannot be disordered by either quantum or thermal fluctuations. Equivalently, when these theories are studied on a circle, no matter how small or large the circle is, the ground state is non-trivial.

We develop new techniques to compute five-point correlation functions from IIB supergravity on AdS 5 × S 5. Our methods rely entirely on symmetry and general consistency conditions, and eschew detailed knowledge of the supergravity effective action. We demonstrate our methods by computing the five-point function of the 20 operator, which is the superconformal primary of the stress tensor multiplet. We also develop systematic methods to compute the five-point conformal blocks in series expansions. Using the explicit expressions of the conformal blocks, we perform a Euclidean OPE analysis of the 20 five-point function. We find expected agreement with non-renormalized quantities and also extract new CFT data at strong coupling.