Conformal Partial Wave Research Articles

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field ε with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the “soft mode” ε, turning them into a simple diagrammatic perturbation theory at large central charge.Our second (equivalent) approach concerns a theory of reparametrization modes, generalizing previous studies in the context of the Schwarzian theory and two-dimensional CFTs. Due to the conformal anomaly in even dimensions, gauge modes of the conformal group acquire an action and are shown to exhibit the same dynamics as the soft mode ε that encodes the physics of the stress tensor shadow. We discuss the calculation of the conformal partial waves or the conformal blocks using our effective field theory. The separation of conformal blocks from shadow blocks is related to gauging of certain symmetries in our effective field theory of the soft mode.These connections explain and generalize various relations between conformal blocks, shadow operators, kinematic space, and reparametrization modes. As an application we study thermal physics in higher dimensions and argue that the theory of reparametrization modes captures the physics of quantum chaos in Rindler space. This is also supported by the observation of the pole skipping phenomenon in the conformal energy-energy two-point function on Rindler space.

Celestial amplitudes: conformal partial waves and soft limits

Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin-transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.

Analytic Euclidean bootstrap

We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading frac{1}{varDelta } tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large ∆ expansion works very well already for small ∆ ∼ 1. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.

Derivation of the Holographic Dual of a Planar Conformal Field Theory in 4D.

We present the first-principles derivation of a weak-strong duality between the fishnet theory in four dimensions and a discretized stringlike model living in five dimensions. At strong coupling, the dual description becomes classical and we demonstrate explicitly the classical integrability of the model. We test our results by reproducing the strong coupling limit of the four-point correlator computed before nonperturbatively from the conformal partial wave expansion. Because of the extreme simplicity of our model, it could provide an ideal playground for holography with no supersymmetry. Furthermore, since the fishnet model and N=4 super Yang-Mills theory are continuously linked, our consideration could shed light on the derivation of AdS/CFT for the latter. For simplicity, in this Letter we restrict our considerations to a large subset of all states.

Recursion relations in Witten diagrams and conformal partial waves

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.

D-dimensional SYK, AdS loops, and 6j symbols

We study the 6j symbol for the conformal group, and its appearance in three seemingly unrelated contexts: the SYK model, conformal representation theory, and perturbative amplitudes in AdS. The contribution of the planar Feynman diagrams to the three-point function of the bilinear singlets in SYK is shown to be a 6j symbol. We generalize the computation of these and other Feynman diagrams to d dimensions. The 6j symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula. We provide closed-form expressions for 6j symbols in d = 1, 2, 4. In AdS, we show that the 6j symbol is the Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging the doubletrace OPE data. Finally, we consider one-loop diagrams in AdS with internal scalars and external spinning operators, and show that the triangle diagram is a 6j symbol, while one-loop n-gon diagrams are built out of 6j symbols.

Scalar blocks as gravitational Wilson networks

In this paper we continue to develop further our prescription [arXiv:1602.02962] to holographically compute the conformal partial waves of CFT correlation functions using the gravitational open Wilson network operators in the bulk. In particular, we demonstrate how to implement it to compute four-point scalar partial waves in general dimension. In the process we introduce the concept of OPE modules, that helps us simplify the computations. Our result for scalar partial waves is naturally given in terms of the Gegenbauer polynomials. We also provide a simpler proof of a previously known recursion relation for the even dimensional CFT partial waves, which naturally leads us to an odd dimensional counterpart.

Anomalous dimensions from crossing kernels

In this note we consider the problem of extracting the corrections to CFT data induced by the exchange of a primary operator and its descendents in the crossed channel. We show how those corrections which are analytic in spin can be systematically extracted from crossing kernels. To this end, we underline a connection between: Wilson polynomials (which naturally appear when considering the crossing kernels given recently in arXiv:1804.09334), the spectral integral in the conformal partial wave expansion, and Wilson functions. Using this connection, we determine closed form expressions for the OPE data when the external operators in 4pt correlation functions have spins J1-J2-0-0, in particular the anomalous dimensions of double-twist operators of the type {left[{mathcal{O}}_{J_1}{mathcal{O}}_{J_2}right]}_{n,ell } in d dimensions and for both leading (n = 0) and sub-leading (n ≠ 0) twist. The OPE data are expressed in terms of Wilson functions, which naturally appear as a spectral integral of a Wilson polynomial. As a consequence, our expressions are manifestly analytic in spin and are valid up to finite spin. We present some applications to CFTs with slightly broken higher-spin symmetry. The Mellin Barnes integral representation for 6j symbols of the conformal group in general d and its relation with the crossing kernels are also discussed.

AdS weight shifting operators

We construct a new class of differential operators that naturally act on AdS harmonic functions. These are weight shifting operators that change the spin and dimension of AdS representations. Together with CFT weight shifting operators, the new operators obey crossing equations that relate distinct representations of the conformal group. We apply our findings to the computation of Witten diagrams, focusing on the particular case of cubic interactions and on massive, symmetric and traceless fields. In particular we show that tree level 4-point Witten diagrams with arbitrary spins, both in the external fields and in the exchanged field, can be reduced to the action of weight shifting operators on similar 4-point Witten diagrams where all fields are scalars. We also show how to obtain the conformal partial wave expansion of these diagrams using the new set of operators. In the case of 1-loop diagrams with cubic couplings we show how to reduce them to similar 1-loop diagrams with scalar fields except for a single external spinning field (which must be a scalar in the case of a two-point diagram). As a bonus, we provide new CFT and AdS weight shifting operators for mixed-symmetry tensors.

Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

AbstractWe study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)‐conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed‐form expressions for crossing kernels of spinning CPWs in terms of the hypergeometric function 4F3. We consider numerous examples involving both arbitrary internal and external spins, and for both leading and sub‐leading twist operators. As an application, working in general d we extract new results for anomalous dimensions of double‐trace operators induced by double‐trace deformations constructed from single‐trace operators of generic twist and integer spin. In particular, we extract the anomalous dimensions of double‐trace operators with a single‐trace operator of integer spin J.

Fermions in geodesic Witten diagrams

We develop the embedding formalism for odd dimensional Dirac spinors in AdS and apply it to the (geodesic) Witten diagrams including fermionic degrees of freedom. We first show that the geodesic Witten diagram (GWD) with fermion exchange is equivalent to the conformal partial waves associated with the spin one-half primary field. Then, we explicitly demonstrate the GWD decomposition of the Witten diagram including the fermion exchange with the aid of the split representation. The geodesic representation of CPW indeed gives the useful basis for computing the Witten diagrams.

Multichannel conformal blocks for scattering amplitudes

By performing resummation of small fermion–antifermion pairs within the pentagon form factor program to scattering amplitudes in planar N=4 superYang–Mills theory, we construct multichannel conformal blocks within the flux-tube picture for N-sided NMHV polygons. This procedure is equivalent to summation of descendants of conformal primaries in the OPE framework. The resulting conformal partial waves are determined by multivariable hypergeometric series of Lauricella–Saran type.

Double-trace deformations of conformal correlations

Large N conformal field theories often admit unitary renormalization group flows triggered by double-trace deformations. We compute the change in scalar four-point functions under double-trace flow, to leading order in 1/N. This has a simple dual in AdS, where the flow is implemented by a change of boundary conditions, and provides a physical interpretation of single-valued conformal partial waves. We extract the change in the conformal dimensions and three-point coefficients of infinite families of double-trace composite operators. Some of these quantities are found to be sign-definite under double-trace flow. As an application, we derive anomalous dimensions of spinning double-trace operators comprised of non-singlet constituents in the O(N) vector model.

Geodesic Witten diagrams with antisymmetric tensor exchange

We show the AdS/CFT correspondence between the conformal partial wave and the geodesic Witten diagram with anti-symmetric exchange. To this end, we introduce the embedding space formalism for anti-symmetric fields in AdS. Then we prove that the geodesic Witten diagram satisfies the conformal Casimir equation and the appropriate boundary condition. Furthermore, we discuss the connection between the geodesic Witten diagram and the shadow formalism by using the split representation of harmonic function for anti-symmetric fields. We also discuss the 3pt geodesic Witten diagrams and its extension to the mixed-symmetric tensors.

DS/CFT and the operator product expansion

Global conformal invariance determines the form of two- and three-point functions of quasiprimary operators in a conformal field theory and generates nontrivial relations between terms in the operator product expansion. These ideas are generalized to the principal and complementary series representations, which play an important role in the conjectured dS/CFT correspondence. The conformal partial wave expansions are constructed for these representations, which in turn determine the operator product expansion. This leads us to conclude that conformal field theories containing such representations have essential singularities, so they cannot be realized as conventional field theories.

Rényi mutual information for a free scalar field in even dimensions

We compute the R\'enyi mutual information of two disjoint spheres in free massless scalar theory in even dimensions higher than two. The spherical twist operator in a conformal field theory can be expanded into the sum of local primary operators and their descendants. We analyze the primary operators in the replicated scalar theory and find the ones of the fewest dimensions and spins. We study the one-point function of these operators in the conical geometry and obtain their expansion coefficients in the OPE of spherical twist operators. We show that the R\'enyi mutual information can be expressed in terms of the conformal partial waves. We compute explicitly the R\'enyi mutual information up to order $z^d$, where $z$ is the cross ratio and $d$ is the spacetime dimension.

Spinning Witten diagrams

We develop a systematic framework to compute the conformal partial wave expansions (CPWEs) of tree-level four-point Witten diagrams with totally symmetric external fields of arbitrary mass and integer spin in AdSd+1. As an intermediate step, we identify convenient bases of three-point bulk and boundary structures to invert linear map between spinning three-point conformal structures and spinning cubic couplings in AdS. Given a CFTd, this provides the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdSd+1. Employing this framework, we determine the CPWE of a generic four-point exchange Witten diagram with spinning exchanged field. As a concrete application, we compute all four-point exchange Witten diagrams in the type A higher-spin gauge theory on AdSd+1, which is conjectured to be dual to the free scalar O (N ) model.

Geodesic diagrams, gravitational interactions & OPE structures

We give a systematic procedure to evaluate conformal partial waves involving symmetric tensors for an arbitrary CFTd using geodesic Witten diagrams in AdSd+1. Using this procedure we discuss how to draw a line between the tensor structures in the CFT and cubic interactions in AdS. We contrast this map to known results using three-point Witten diagrams: the maps obtained via volume versus geodesic integrals differ. Despite these differences, we show how to decompose four-point exchange Witten diagrams in terms of geodesic diagrams, and we discuss the product expansion of local bulk fields in AdS.

Anatomy of geodesic Witten diagrams

We revisit the so-called “Geodesic Witten Diagrams” (GWDs) [1], proposed to be the holographic dual configuration of scalar conformal partial waves, from the perspectives of CFT operator product expansions. To this end, we explicitly consider three point GWDs which are natural building blocks of all possible four point GWDs, discuss their gluing procedure through integration over spectral parameter, and this leads us to a direct identification with the integral representation of CFT conformal partial waves. As a main application of this general construction, we consider the holographic dual of the conformal partial waves for external primary operators with spins. Moreover, we consider the closely related “split representation” for the bulk to bulk spinning propagator, to demonstrate how ordinary scalar Witten diagram with arbitrary spin exchange, can be systematically decomposed into scalar GWDs. We also discuss how to generalize to spinning cases.

Geodesic Witten diagrams with an external spinning field

We explore AdS/CFT correspondence between geodesic Witten diagrams and conformal blocks (conformal partial waves) with an external symmetric traceless tensor field. We derive an expression for the conformal partial wave with an external spin-1 field and show that this expression is equivalent to the amplitude of the geodesic Witten diagram. We also show the equivalence by using conformal Casimir equation in embedding formalism. Furthermore, we extend the construction of the amplitude of the geodesic Witten diagram to an external arbitrary symmetric traceless tensor field. We show our construction agrees with the known result of the conformal partial waves.