Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.
We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complement…
This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT$_{1}$ dual to AdS$_{2}$ black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS$_{2}$ black hole and projective unitary representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. We derive novel recurrence …
We study real-time correlators in the warped AdS/CFT correspondence. We apply the prescription used in the usual AdS/CFT correspondence and obtain the retarded Green's functions for the scalar and vector fields in the spacelike warped and the null warped black hole backgrounds. We find that the retarded Green's functions and the cross sections are well consistent with the predictions from dual CFT. Our results not only support strongly the conjectured warped AdS/CFT correspondence, but also show that the usual relativistic AdS/CFT prescription of obtaining the real-time correlators remain effe…
We calculate all components of thermal R-current correlators from AdS/CFT correspondence for non-zero momentum and energy. In zero momentum limit, we find an analytic expression for the components Gxx(Gyy). The dielectric function of strong coupling is also presented and compared with that in weak coupling.
We consider the problem of bounding CFT correlators on the Euclidean section. By reformulating the question as an optimization problem, we construct functionals numerically which determine upper and lower bounds on correlators under several circumstances. A useful outcome of our analysis is that the gap maximization bootstrap problem can be reproduced by a numerically easier optimization problem. We find that the 3d Ising spin correlator takes the minimal possible allowed values on the Euclidean section. Turning to the maximization problem we find that for d > 2 there are gap-independent maxim…
We present a simple Lie-algebraic approach to momentum-space two-point functions of two-dimensional conformal field theory at finite temperature dual to the BTZ black hole. Making use of the real-time prescription of AdS/CFT correspondence and ladder equations of the Lie algebra $\mathfrak{so}(2,2) \cong \mathfrak{sl}(2,\mathbb{R})_{L} \oplus \mathfrak{sl}(2,\mathbb{R})_{R}$, we show that the finite-temperature two-point functions in momentum space satisfy linear recurrence relations with respect to the left and right momenta. These recurrence relations are exactly solvable and completely dete…
Extending earlier results by Paulos, we discuss further the use of the embedding formalism and Mellin transform in the calculation of tree-level correlators of scalar and vector fields in AdS/CFT. We present an iterative procedure that builds amplitudes by sewing together lower-point off-shell diagrams. Both scalar and vector correlators are shown to be given in terms of Mellin amplitudes. We apply the procedure to the explicit calculation of three-, four- and five-point correlators.
We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fi…
We study one and two point functions of conformal field theories on spaces of maximal symmetry with and without boundaries and investigate their spectral representations. Integral transforms are found, relating the spectral decomposition to renormalized position space correlators. Several applications are presented, including the holographic boundary CFTs as well as spacelike boundary CFTs, which provide realizations of the pseudo-conformal universe.
We discuss the case of correlators in CFT made of pure contact terms, without a corresponding bare part. We show two examples. The first is provided by the conformal limits of a free massive fermion theory in 3d. We show that the (conserved) current correlators are in one-to-one correspondence with the terms of the 3d gauge CS action. The second is the Pontryagin trace anomaly. The corresponding 3-point correlator is nonvanishing even though the corresponding untraced correlator vanishes.
Conformal field theory (CFT) dispersion relations reconstruct correlators in terms of their double discontinuity. When applied to the crossing equation, such dispersive transforms lead to sum rules that suppress the double-twist sector of the spectrum and enjoy positivity properties at large twist. In this paper, we construct dispersive CFT functionals for correlators of unequal scalar operators in position- and Mellin-space. We then evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter s…
We show that general parity-violating 3d conformal field theories show a double copy structure for momentum space 3-point functions of conserved currents, stress tensor and marginal scalar operators. Splitting up the CFT correlator into two parts - called homogeneous and non-homogeneous - we show that double copy relations exist for each part separately. We arrive at similar conclusions regarding double copy structures using tree-level correlators of massless fields in $dS_4$. We also discuss the flat space limit of these correlators. We further extend the double copy analysis to correlators i…
We study real-time correlators in the Kerr/CFT correspondence. The near-horizon geometry of extreme and near-extremal Kerr black holes can be taken as warped spacetimes with the warping factors depending on the transverse radius and the angular variable. By considering the Kerr/CFT correspondence as a generalized warped AdS/CFT correspondence, we show how to compute the CFT real-time correlation functions. We find that the retarded Green's functions, the cross sections and the quasi-normal modes are all in perfect match with the predictions of the dual CFT. The same analysis is also generalize…
We consider the momentum-space 3-point correlators of currents, stress tensors and marginal scalar operators in general odd-dimensional conformal field theories. We show that the flat space limit of these correlators is spanned by gauge and gravitational scattering amplitudes in one higher dimension which are related by a double copy. Moreover, we recast three-dimensional CFT correlators in terms of tree-level Feynman diagrams without energy conservation, suggesting double copy structure beyond the flat space limit.
We extend to chiral N=4 operators the holographic agreement recently found between correlators of the symmetric orbifold of M^4 at large N and type IIB strings propagating in AdS(3) x S^3 x M^4, where M^4=T^4 or K3. We also present expressions for some bulk correlators not yet computed in the boundary.
We systematically study correlators of a generic conformal field theory with a global O(2) symmetry in a sector of large global charge. We focus in particular on three- and four-point correlators with conserved current insertions sandwiched between spinful excited states corresponding to phonons over the large-charge vacuum. We also discuss loop corrections to the scaling dimensions and observe multiple logarithms in even dimensions.
In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory ($χ$CFT$_4$) arising as a double scaling limit of the $γ$-deformed $\mathcal{N}=4$ SYM theory. In the planar (t'Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice "fishnet" and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with $SO(1,5)$ symmetry. In this paper we explain in det…
N-point correlation functions of conserved currents and weight-two scalar operators of the three-dimensional free fermion vector model are found as invariants of the higher-spin symmetry in four-dimensional AdS. These are the correlators of the unbroken Vasiliev higher-spin theory. The results extend the recent work arXiv:1210.7963 and are complementary to arXiv:1301.3123 where the correlators were computed entirely on the boundary.
We show that next-to-extremal correlators of chiral primary operators in N=4 SYM theory do not receive quantum corrections to first order in perturbation theory. Furthermore we consider next-to-extremal correlators within AdS supergravity. Here the exchange diagrams contributing to these correlators yield results of the same free-field form as obtained within field theory. This suggests that quantum corrections vanish at strong coupling as well.
We show that the scaling dimensions of lowest operators in conformal field theories (CFTs) can be isolated in small and closed regions from single correlator bootstrap. We find the conserved currents play crucial roles in bootstrapping the crossing equation. By imposing a mild gap between the scaling dimensions of the conserved current and its next operator, the scaling dimensions of lowest operators are forced to lie in small isolated regions, i.e., these CFTs can be almost fixed by few lowest operators in certain channels. For CFTs with extended supersymmetry, the single correlator crossing …
The AdS/CFT correspondence predicts a phase transition in Wilson loop correlators in the strong coupling N=4, D=4 SYM theory which arises due to instability of the classical string stretched between the loops. We study this transition in detail by solving equations of motion for the string in the particular case of two circular Wilson loops. The transition is argued to be smoothened at finite `t Hooft coupling by fluctuations of the string world sheet and to be promoted to a sharp crossover. Some general comments about Wilson loop correlators in gauge theories are made.
We study the two circular Wilson loop correlator in which one is of anti-symmetric representation, while the other is of fundamental representation in 4-dimensional ${\cal N}=4$ super Yang-Mills theory. This correlator has a good AdS dual, which is a system of a D5-brane and a fundamental string. We calculated the on-shell action of the string, and clarified the Gross-Ooguri transition in this correlator. Some limiting cases are also examined.
Two-dimensional conformal field theories at large central charge and with a sufficiently sparse spectrum of light states have been shown to exhibit universal thermodynamics. This thermodynamics matches that of AdS$_3$ gravity, with a Hawking-Page transition between thermal AdS and the BTZ black hole. We extend these results to correlation functions of light operators. Upon making some additional assumptions, such as large $c$ factorization of correlators, we establish that the thermal AdS and BTZ solutions emerge as the universal backgrounds for the computation of correlators. In particular, W…
By solving the equations of motion of massive $p$-form potential in Anti-de-Sitter space and using the $AdS/CFT$ correspondence of Maldacena, the generating functional of two-point correlation functions of the currents is obtained. When the mass parameter vanishes the result agrees with the known massless case.
We study conformal twist field four-point functions on a $\mathbb Z_N$ orbifold. We examine in detail the case $N=3$ and analyze theories obtained by replicated $N$-times a minimal model with central charge $c<1$. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge $Nc$ is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density m…
We continue our study of string correlators on Euclidean $\text{AdS}_3$ with pure NS-NS flux. The worldsheet and spacetime correlators have a rich analytic structure, which we analyse completely for genus 0 four-point functions. We show that correlators exhibit a simple behaviour near their singularities. The spacetime correlators are meromorphic functions in the $\mathrm{SL}(2,\mathbb{R})$-spins, whose pole structure is shown to agree with the prediction of a recent proposal for the dual $\text{CFT}_2$. Moreover, we also compute the residues of the spacetime correlators for some of the poles …
We consider general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group, and generalize the Lunin-Mathur covering space technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. We work two examples, one using a simple bosonic CFT, and one using the …
We compute the exact retarded Green's functions in thermal $CFT$s with chemical potential and angular momenta using holography respectively. We consider the field equations satisfied by the quasi-normal modes in both charged and rotating black holes in $AdS$ spacetime and mapped them to the Heun equations by appropriate changes of variables. The AGT correspondence allows us to find the connection formulae among the solutions of the Heun equations near different singularities by using the crossing relations of the five-point correlators in the Liouville $CFT$. The connection formulae associated…
Recently exact agreement has been found between three-point correlators of chiral operators computed in string theory on AdS_3 x S^3 x T^4 with NS-NS flux and those computed in the symmetric orbifold CFT. However, it has also been shown that these correlators disagree with those computed in supergravity, under any identification of single particle operators which respects the symmetries. In this note we resolve this disagreement: the key point is that mixings with multi-particle operators are not suppressed even at large N in extremal correlators. Allowing for such mixings, orbifold/string the…
We consider current-current correlators in 4d $\N =1$ SCFTs, and also 3d $\N =2$ SCFTs, in connection with AdS/CFT geometry. The superconformal $U(1)_R$ symmetry of the SCFT has the distinguishing property that, among all possibilities, it minimizes the coefficient, $τ_{RR}$ of its two-point function. We show that the geometric Z-minimization condition of Martelli, Sparks, and Yau precisely implements $τ_{RR}$ minimization. This gives a physical proof that Z-minimization in geometry indeed correctly determines the superconformal R-charges of the field theory dual. We further discuss and compar…
In this paper, we study the real-time correlators in Kerr/CFT, in the low frequency limit of generic non-extremal Kerr(-Newman) black holes. From the low frequency scattering of Kerr-Newman black holes, we show that for the uncharged scalar scattering, there exists hidden conformal symmetry on the solution space. Similar to Kerr case, this suggests that the Kerr-Newman black hole is dual to a two-dimensional CFT with central charges $c_L=c_R=12J$ and temperatures $T_L=\frac{(r_++r_-)-Q^2/M}{4πa}, T_R=\frac{r_+-r_-}{4πa}$. Using the Minkowski prescription, we compute the real-time correlators o…
A previous calculation of the phase transition in the Wilson loop correlator in the zero temperature AdS/CFT correspondence is extended to the case where the loops are concentric circles of unequal radii. This phase transition occurs due to the instability of the classical string stretched between the loops. We compute the string action and its expansion in the distance h between the loops for small h. We also find that the connected minimal surface is subleading or does not even exist when h=0 and the radii are considerably different. This feature has no analogue in flat space.
We compute three-point correlation functions in the near-extremal, near-horizon region of a Kerr black hole, and compare to the corresponding finite-temperature conformal field theory correlators. For simplicity, we focus on scalar fields dual to operators ${\cal O}_h$ whose conformal dimensions obey $h_3=h_1+h_2$, which we name \emph{extremal} in analogy with the classic $AdS_5 \times S^5$ three-point function in the literature. For such extremal correlators we find perfect agreement with the conformal field theory side, provided that the coupling of the cubic interaction contains a vanishing…
We calculate, for the first time, three-point correlation functions involving "heavy" operators in the Schrodinger/null-dipole CFT correspondence at strong coupling. In particular, we focus on the three-point functions of the dilaton modes and two "heavy" operators. The heavy states are dual to the single spin and dyonic magnon, the single spin and dyonic spike solutions or to two novel string solutions which do not have an undeformed counterpart. Our results provide the leading term of the correlators in the large $λ$ expansion and are in perfect agreement with the form of the correlator dict…
The BPS correlators of the symmetric product orbifold $\text{Sym}_N(\mathbb{T}^4)$ are reproduced from the dual worldsheet theory describing strings on $\text{AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with minimal ($k=1$) NS-NS flux. More specifically, we show that the worldsheet duals of the symmetric orbifold BPS states can be identified with their lift to the covering surface, thereby making the matching of the correlators essentially manifest. We also argue that the argument can be generalised to arbitrary descendants, using suitable DDF operators on the worldsheet.
We use large spin perturbation theory and the Lorentzian inversion formula to compute order-$\varepsilon$ corrections to mixed correlators in the $O(n)$ Wilson-Fisher CFT in $4 - \varepsilon$ dimensions. In particular, we find the scaling dimensions and averaged OPE coefficients appearing in all correlators involving $φ$ and $φ^2$, for $φ^2$ in both the singlet and symmetric traceless representations of $O(n)$. We extend some computations to the next order, and find order-$\varepsilon^2$ data for a number of quantities for the Ising case at $n = 1$. Along the way, we discuss several interestin…
The AdS_3/CFT_2 correspondence is checked beyond the supergravity approximation by comparing correlation functions. To this end we calculate 2- and 3-point functions on the sphere of certain chiral primary operators for strings on AdS_3 x S^3 x T^4. These results are then compared with the corresponding amplitudes in the dual 2-dimensional conformal field theory. In the limit of small string coupling, where the sphere diagrams dominate the string perturbation series, beautiful agreement is found.
To O(1/N) we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expe…
We consider correlation functions of the stress-tensor or a conserved current in AdS_{d+1}/CFT_d computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the BCFW-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including d=3. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the (d+1)-dimensional flat-space amplitude of gravitons …
We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT. Specifically, we consider correlators of descendant operators constructed using both the full Virasoro generators $L_{m}$ and fractional Virasoro generators $\ell_{m/n_i}$. Using covering space techniques, we show that correlators of descendants may be written entirely in terms of correlators of ancestors, and further that the appropriate set of ancestors are those operators that lift to conformal primaries on the cover. We argue that the covering space data should cancel out in s…
In this article we probe the proposed holographic duality between $T\bar{T}$ deformed two dimensional conformal field theory and the gravity theory of $\rm AdS_3$ with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $T\bar{T}$ CFT in a Euclidean plane and in a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in $\rm AdS_3$ with the corresponding cutoff surface, given a hologra…
We study non-Gaussian bulk 2d CFTs in AdS$_2$ using boundary CFT techniques and recent results in JT/Schwarzian gravity. We highlight the constraints on the operator content of a theory imposed by the boundary conditions by examining the relation between correlator coefficients and OPE coefficients in the presence of a boundary. We then calculate bulk and boundary correlators for various boundary conditions. Schwarzian techniques are used to calculate gravitational correlators perturbatively in $1/c$.
We demonstrate how three-point correlation functions of the free scalar U(N) model involving two scalar operators and one spin-$s$ conserved current organize themselves into corresponding AdS amplitudes involving two scalar and one spin-$s$ bulk to boundary propagators, coupled via the bulk gauge invariant interaction vertex. Our analysis relies on the general program advocated in hep-th/0308184 and some features of the embedding space formalism also play an important role.
Unitary conformal field theories (CFTs) are believed to have positive (non-negative) energy correlators. Energy correlators are universal observables in higher-dimensional CFTs built out of integrated Wightman functions of the stress-energy tensor. We analyze energy correlators in parity invariant four-dimensional CFTs. The goal is to use the positivity of energy correlators to further constrain unitary CFTs. It is known that the positivity of the simplest one-point energy correlator implies that 1/3 <= a/c <= 31/18 where a and c are the Weyl anomaly coefficients. We use the positivity of high…
We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinni…
All correlation functions of conserved currents of the CFT that is dual to unbroken Vasiliev theory are found as invariants of higher-spin symmetry in the bulk of AdS. The conformal and higher-spin symmetry of the correlators as well as the conservation of currents are manifest, which also provides a direct link between the Maldacena-Zhiboedov result and higher-spin symmetries. Our method is in the spirit of AdS/CFT, though we never take any boundary limit or compute any bulk integrals. Boundary-to-bulk propagators are shown to exhibit an algebraic structure, living at the boundary of SpH(4), …
