Search

Abstract We perform a systematic bootstrap analysis of four-point one-loop Mellin amplitudes for super gluons in AdS5 × S3 with arbitrary Kaluza-Klein weights. The analysis produces the general expressions for these amplitudes at extremalities two and three, as well as analytic results for many other special cases. From these results we observe remarkable simplicity. We find that the Mellin amplitudes always contain only simultaneous poles in two Mellin-Mandelstam variables, extending a previous observation in the simplest case with the lowest Kaluza-Klein weights. Moreover, we discover a substantial extension of the implication of the eight-dimensional hidden conformal symmetry, which goes far beyond the Mellin poles associated with the leading logarithmic singularities. This leaves only a small finite set of poles which can be determined on a case-by-case basis from the contributions of protected operators in the OPE.

Abstract We study three-point correlation functions of scalar operators in conformal field theories with boundaries and interfaces. We focus on two cases where there are one bulk and two boundary operators (B∂∂), or two bulk and one boundary operators (BB∂). We perform a detailed analysis of the conformal blocks in different OPE channels. In particular, we obtain the bulk channel conformal blocks of the BB∂ three-point functions for arbitrary exchanged spins in a series expansion with respect to the radial coordinates. We also study examples of such three-point functions in the simplest holographic dual where the AdS d+1 space contains a brane filling an AdS d subspace. Such a setup arises in top-down models with probe branes and is also relevant for the functional approach to boundary and interface CFT correlators. We systematically study the Witten diagrams in this setup both in position space and in Mellin space. We also discuss in detail how to decompose these Witten diagrams into conformal blocks.

Abstract We continue to explore the bootstrap approach to five-point correlation functions for IIB supergravity on AdS 5 × S 5. Building on the result of [1], we develop an improved algorithm that allows us to more efficiently compute correlators of higher Kaluza-Klein modes. The new method uses only factorization and a superconformal twist, and is entirely within Mellin space where the analytic structure of holographic correlators is simpler. Using this method, we obtain in a closed form all five-point functions of the form 〈pp222〉, extending the earlier result for p = 2. As a byproduct of our analysis, we also obtain explicit results for spinning four-point functions of higher Kaluza-Klein modes.

Abstract We carry out a bootstrap study of four-point correlators in 4d N $$ \mathcal{N} $$ = 2 SCFTs which are dual to super Yang-Mills on AdS 5 × S 3. We focus on the simplest 1 2 $$ \frac{1}{2} $$ -BPS operators which correspond to the super gluons in the massless current multiplet. Our computation is based on an ansatz in position space which is inspired by a hidden symmetry structure manifest in the leading terms of the Lorentzian singularities of the correlators. By using other consistency conditions, we completely fix the super gluon correlators at one and two loops in the bulk genus expansion, up to possible counterterms. Our results reveal a number of interesting properties enriched by the color structures. In particular, the implication of hidden conformal symmetry on the full super gluon reduced correlator exhibits an analogous pattern as in the AdS 5 × S 5 supergravity correlators recently computed up to two loops.

Abstract We study four-point functions of spinning operators in the flavor current multiplet in four dimensional N $$ \mathcal{N} $$ = 2 SCFTs, using superspace techniques. In particular we explicitly construct the differential operators relating the different components of the super-correlator. As a byproduct of our analysis, we report the computation of the four-point amplitudes of gluons in bosonic Yang-Mills theories on AdS5 and we give evidence of an AdS double copy relation between the gluon amplitude and its gravitational counterpart.

Abstract We initiate the study of bound state scattering in AdS space at the level of Witten diagrams. For concreteness, we focus on the case with only scalar fields and analyze several basic diagrams which more general diagrams reduce to. We obtain closed form expressions for their Mellin amplitudes with arbitrary conformal dimensions, which exhibit interesting behavior. In particular, we observe that certain tree-level bound state Witten diagrams have the same structure as loop diagrams in AdS.

Abstract We initiate the study of one-loop gluon amplitudes in AdS space. These amplitudes were recently computed at tree level for a variety of backgrounds of the form AdS d+1 × S 3. For concreteness, we compute the one-loop correction to the massless gluon amplitude on AdS 5 ×S 3, which corresponds to the four-point correlator of the flavor current multiplet in the dual 4d N $$ \mathcal{N} $$ = 2 SCFT. This requires solving a mixing problem that involves tree-level amplitudes of arbitrarily massive Kaluza-Klein modes. The final answer has the same color structure as in flat space but the dependence on Mandelstam variables is more complicated, with logarithms replaced by polygamma functions.

Abstract We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

Abstract We present a systematic study of holographic correlators in a vast array of SCFTs with non-maximal superconformal symmetry. These theories include 4d N $$ \mathcal{N} $$ = 2 SCFTs from D3-branes near F-theory singularities, 5d Seiberg exceptional theories and 6d E-string theory, as well as 3d and 4d phenomenological models with probe flavor branes. We consider current multiplets and their generalizations with higher weights, dual to massless and massive super gluons in the bulk. At leading order in the inverse central charge expansion, connected four-point functions of these operators correspond to tree-level gluon scattering amplitudes in AdS. We show that all such tree-level four-point amplitudes in all these theories are fully fixed by symmetries and consistency conditions and explicitly construct them. Our results encode a wealth of SCFT data and exhibit various interesting emergent structures. These include Parisi-Sourlas-like dimensional reductions, hidden conformal symmetry and an AdS version of the color-kinematic duality.

Abstract Recently four-point holographic correlators with arbitrary external BPS operators were constructively derived in [1, 2] at tree-level for maximally superconformal theories. In this paper, we capitalize on these theoretical data, and perform a detailed study of their analytic properties. We point out that these maximally supersymmetric holographic correlators exhibit a hidden dimensional reduction structure à la Parisi and Sourlas. This emergent structure allows the correlators to be compactly expressed in terms of only scalar exchange diagrams in a dimensionally reduced spacetime, where formally both the AdS and the sphere factors have four dimensions less. We also demonstrate the superconformal properties of holographic correlators under the chiral algebra and topological twistings. For AdS5 × S 5 and AdS7 × S 4, we obtain closed form expressions for the meromorphic twisted correlators from the maximally R-symmetry violating limit of the holographic correlators. The results are compared with independent field theory computations in 4d N $$ \mathcal{N} $$ = 4 SYM and the 6d (2, 0) theory, finding perfect agreement. For AdS4 × S 7, we focus on an infinite family of near-extremal four-point correlators, and extract various protected OPE coefficients from supergravity. These OPE coefficients provide new holographic predictions to be matched by future supersymmetric localization calculations. In deriving these results, we also develop many technical tools which should have broader applicability beyond studying holographic correlators.

Abstract We demonstrate the simplicity of AdS 5 × S 5 IIB supergravity at one loop level, by studying non-planar holographic four-point correlators in Mellin space. We develop a systematic algorithm for constructing one-loop Mellin amplitudes from the tree-level data, and obtain a simple closed form answer for the O 2 SG O 2 SG O p SG O p SG $$ \left\langle {\mathcal{O}}_2^{SG}{\mathcal{O}}_2^{SG}{\mathcal{O}}_p^{SG}{\mathcal{O}}_p^{SG}\right\rangle $$ correlators. The structure of this expression is remarkably simple, containing only simultaneous poles in the Mellin variables. We also study the flat space limit of the Mellin amplitudes, which reproduces precisely the IIB supergravity one-loop amplitude in ten dimensions. Our results provide nontrivial evidence for the persistence of the hidden conformal symmetry at one loop.

Abstract Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in d and d − 2 dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.

Abstract Surface operators in the 6d (2,0) theory at large N have a holographic description in terms of M2 branes probing the AdS7 × S 4 M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an SO(2, 2) subgroup of d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS3 induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the defect CFT data associated to the surface operator. The spectrum of transverse fluctuations of the M2 brane is found to be in one-to-one correspondence with a protected multiplet of operator insertions on the surface, which includes the displacement operator. We compute the one-loop determinants of fluctuations of the M2 brane, and extract the conformal anomaly coefficient of the spherical surface to order N 0. We also briefly discuss the RG flow from the non-supersymmetric to the 1/2-BPS defect operator, and its consistency with a “b-theorem” for the defect CFT. Starting with the M2 brane action, we then use AdS3 Witten diagrams to compute the 4-point functions of the elementary bosonic insertions on the surface operator, and extract some of the defect CFT data from the OPE. The 4-point function is shown to satisfy superconformal Ward identities, and we discuss a related sub- sector of “twisted” scalar insertions, whose correlation functions are constrained by the residual superconformal symmetry.

Abstract We develop an analytic approach to Boundary Conformal Field Theory (BCFT), focussing on the two-point function of a general pair of scalar primary operators. The resulting crossing equation can be thought of as a vector equation in an infinite-dimensional space V $$ \mathcal{V} $$ of analytic functions of a single complex variable. We argue that in a unitary theory, functions in V $$ \mathcal{V} $$ satisfy a boundedness condition in the Regge limit. We identify a useful basis for V $$ \mathcal{V} $$ , consisting of bulk and boundary conformal blocks with scaling dimensions which appear in OPEs of the mean field theory correlator. Our main achievement is an explicit expression for the action of the dual basis (the basis of linear functionals on V $$ \mathcal{V} $$ ) on an arbitrary conformal block. The practical merit of our basis is that it trivializes the study of perturbations around mean field theory. Our results are equivalent to a BCFT version of the Polyakov bootstrap. Our derivation of the expressions for the functionals relies on the identification of the Polyakov blocks with (suitably improved) boundary and bulk Witten exchange diagrams in AdS d+1. We also provide another conceptual perspective on the Polyakov block expansion and the associated functionals, by deriving a new Lorentzian OPE inversion formula for BCFT.

Abstract We develop new techniques to compute five-point correlation functions from IIB supergravity on AdS5 × S 5. Our methods rely entirely on symmetry and general con- sistency 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 ex- plicit 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.

Abstract We revisit the calculation of holographic correlators in AdS 3. We develop new methods to evaluate exchange Witten diagrams, resolving some technical difficulties that prevent a straightforward application of the methods used in higher dimensions. We perform detailed calculations in the AdS 3 × S 3 × K 3 background. We find strong evidence that four-point tree-level correlators of KK modes of the tensor multiplets enjoy a hidden 6d conformal symmetry. The correlators can all be packaged into a single generating function, related to the 6d flat space superamplitude. This generalizes an analogous structure found in AdS 5 × S 5 supergravity.

Abstract We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.

Abstract We present a universal treatment for imposing superconformal constraints on Mellin amplitudes for SCFTd 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 four-point 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 $$ \mathcal{N}=8 $$ numerical bootstrap bound at large central charge.

Abstract We extend the Mellin space techniques of [1] for computing holographic four-point correlation functions in maximally superconformal theories to theories with only eight Poincaré supercharges. The one-half BPS operators in these correlators are taken to be the superconformal primary in the Dk $$ \mathcal{D}\left[k\right] $$ multiplet (with k = 2 corresponding to the flavor current multiplet), and transform in the adjoint representation of a flavor group G. Because of the smaller R-symmetry group SU(2), each individual superconformal Ward identity is less powerful. On the other hand, the constraining power is compensated in number by the different flavor channels in the four-point function. As concrete test cases, we study the Seiberg theories in five dimensions and E-string theory in six dimensions at the large N limit. We show that the flavor current multiplet four-point functions are fixed by superconformal symmetry up to two free parameters, which are proportional to the squared OPE coefficients for the flavor current multiplet and the stress tensor multiplet.

Abstract We revisit the calculation of holographic correlators for eleven-dimensional supergravity on AdS 7 × S 4. 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 AdS 5 × S 5 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.

Abstract We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on AdS5 × 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 one-half 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 “on-shell 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.

Abstract We extend the Mellin representation of conformal field theory (CFT) to allow for conformal boundaries and interfaces. We consider the simplest holographic setup dual to an interface CFT — a brane filling an AdS d subspace of AdS d+1 — and perform a systematic study of Witten diagrams in this setup. As a byproduct of our analysis, we show that geodesic Witten diagrams in this geometry reproduce interface CFT d conformal blocks, generalizing the analogous statement for CFTs with no defects.