Single-trace Operators Research Articles

Supercharged AdS3 Holography

Given an asymptotically Anti-de Sitter supergravity solution, one can obtain a microscopic interpretation by identifying the corresponding state in the holographically dual conformal field theory. This is of particular importance for heavy pure states that are candidate black hole microstates. Expectation values of light operators in such heavy CFT states are encoded in the asymptotic expansion of the dual bulk configuration. In the D1-D5 system, large families of heavy pure CFT states have been proposed to be holographically dual to smooth horizonless supergravity solutions. We derive the precision holographic dictionary in a new sector of light operators that are superdescendants of scalar chiral primaries of dimension (1,1). These operators involve the action of the supercharges of the chiral algebra, and they play a central role in the proposed holographic description of recently-constructed supergravity solutions known as “supercharged superstrata”. We resolve the mixing of single-trace and multi-trace operators in the CFT to identify the combinations that are dual to single-particle states in the bulk. We identify the corresponding gauge-invariant combinations of supergravity fields. We use this expanded dictionary to probe the proposed holographic description of supercharged superstrata, finding precise agreement between gravity and CFT.

Four point functions in CFT\u2019s with slightly broken higher spin symmetry

We compute spinning four point functions in the quasi-fermionic three dimensional conformal field theory with slightly broken higher spin symmetry at finite t’Hooft coupling. More concretely, we obtain a formula for leftlangle {j}_s{j}_{tilde{0}}{j}_{tilde{0}}{j}_{tilde{0}}rightrangle , where js is a higher spin current and {j}_{tilde{0}} is the scalar single trace operator. Our procedure consists in writing a plausible ansatz in Mellin space and using crossing, pseudo-conservation and Regge boundedness to fix all undetermined coefficients. Our method can potentially be generalised to compute all spinning four point functions in these theories.

Two-point correlator of chiral primary operators with a Wilson line defect in mathcal{N} = 4 SYM

We study the two-point function of the stress-tensor multiplet of mathcal{N} = 4 SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the 20′ irrep of the mathfrak{so} (6)R R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of mathcal{N} = 4 SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques.

From sources to initial data and back again: on bulk singularities in Euclidean AdS/CFT

A common method to prepare states in AdS/CFT is to perform the Euclidean path integral with sources turned on for single-trace operators. These states can be interpreted as coherent states of the bulk quantum theory associated to Lorentzian initial data on a Cauchy slice. In this paper, we discuss the extent to which arbitrary initial data can be obtained in this way. We show that the initial data must be analytic and define the subset of it that can be prepared by imposing bulk regularity. Turning this around, we show that for generic analytic initial data the corresponding Euclidean section contains singularities coming from delta function sources in the bulk. We propose an interpretation of these singularities as non-perturbative objects in the microscopic theory.

Shocks, superconvergence, and a stringy equivalence principle

We study propagation of a probe particle through a series of closely situated gravitational shocks. We argue that in any UV-complete theory of gravity the result does not depend on the shock ordering — in other words, coincident gravitational shocks commute. Shock commutativity leads to nontrivial constraints on low-energy effective theories. In particular, it excludes non-minimal gravitational couplings unless extra degrees of freedom are judiciously added. In flat space, these constraints are encoded in the vanishing of a certain “superconvergence sum rule.” In AdS, shock commutativity becomes the statement that average null energy (ANEC) operators commute in the dual CFT. We prove commutativity of ANEC operators in any unitary CFT and establish sufficient conditions for commutativity of more general light-ray operators. Superconvergence sum rules on CFT data can be obtained by inserting complete sets of states between light-ray operators. In a planar 4d CFT, these sum rules express frac{a-c}{c} in terms of the OPE data of single-trace operators.

Structure constants in mathcal{N} = 4 SYM at finite coupling as worldsheet g-function

We develop a novel nonperturbative approach to a class of three-point functions in planar mathcal{N} = 4 SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS5×S5. They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coupling; one based on large N collective fields, which provides an example of open-closed-open duality discussed by Gopakumar, and the other based on combinatorics. The results so obtained exhibit a simple determinant structure and indicate that the correlator can be interpreted as a generalization of g-functions in 2d QFT; an overlap between an integrable boundary state and a state corresponding to the single-trace operator. We then determine the boundary state at finite coupling using the symmetry, the crossing equation and the boundary Yang-Baxter equation. With the resulting boundary state, we derive the ground-state g-function based on TBA and conjecture its generalization to other states. This is the first fully nonperturbative proposal for the structure constants of operators of finite length. The results are tested extensively at weak and strong couplings. Finally, we point out that determinant operators can provide better probes of sub-AdS locality than single-trace operators and discuss possible applications.

Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings

In the previous papers, it is pointed out that a supersymmetric double-well matrix model corresponds to a two-dimensional type IIA superstring theory on a Ramond-Ramond background at the level of correlation functions. This was confirmed by agreement between their planar correlation functions. The supersymmetry in the matrix model corresponds to the target space supersymmetry and it is shown to be spontaneously broken by nonperturbative effect. Furthermore, in the matrix model we computed one-point functions of single-trace operators to all order of genus expansion in its double scaling limit. We found that this expansion is stringy and not Borel summable and hence there arises an ambiguity in applying the Borel resummation technique. We confirmed that resurgence works here, namely this ambiguity in perturbative series in a zero-instanton sector is exactly canceled by another ambiguity in a one-instanton sector obtained by instanton calculation. In this paper we extend this analysis and study resurgence structure of the two-point functions of the single trace operators. By using results in the random matrix theory, we derive two-point functions at arbitrary genus and see that the perturbative series in the zero-instanton sector again has an ambiguity. We find that the two-point functions inevitably have logarithmic singularity even at higher genus. In this derivation we obtain a new result of the two-point function expressed by the one-point function at the leading order in the soft-edge scaling limit of the random matrix theory. We also compute an ambiguity in the one-instanton sector by using the Airy kernel, and confirm that ambiguities in both sectors cancel each other at the leading order in the double scaling limit. We thus clarify resurgence structure of the two-point functions in the supersymmetric double-well matrix model.

Colour-twist operators. Part I. Spectrum and wave functions

We introduce a new class of operators in any theory with a ’t Hooft large-N limit that we call colour-twist operators. They are defined by twisting the colour-trace with a global symmetry transformation and are continuously linked to standard, untwisted single-trace operators. In particular, correlation functions between operators that are twisted by an R-symmetry of mathcal{N} = 4 SYM extend those in the γ-deformed theory. The most general deformation also breaks the Lorentz symmetry but preserves integrability in the examples we consider. In this paper, we focus on colour-twist operators in the fishnet model. We exemplify our approach for the simplest colour-twist operators with one and two scalar fields, which we study non-perturbatively using field-theoretical as well as integrability methods, finding a perfect match. We also propose the quantisation condition for the Baxter equation appearing in the integrability calculation in the fishnet model. The results of this paper constitute a crucial step towards building the separation of variable construction for the correlation functions by means of the Quantum Spectral Curve approach.

AdS/CFT unitarity at higher loops: high-energy string scattering

What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large N , 4d CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in 1/N for large, but finite, coupling. In the language of Regge theory, we show Pomeron exchange naturally exponentiates in the 1/N expansion. To do this, we solve the bootstrap equations at tree-level and then use the Lorentzian inversion formula to find the one-loop correlator in the Regge limit. This is a unitarity method for AdS/CFT which can be repeated iteratively to make all orders statements. We also explain under what conditions the tree-level result exponentiates in the 1/N expansion at arbitrary coupling. Finally, we comment on further inelastic effects and show they give subleading contributions at large coupling. As a consistency check, we recover results from bulk Einstein gravity in the limit where all higher-spin particles decouple.

Correlation functions in mathcal{N} = 2 Supersymmetric vector matter Chern-Simons theory

Correlation functions of the higher-spin current operators in large N Chern-Simons theories are important to understand approximate higher-spin symmetries in these theories. Moreover, they also provide stronger checks for conjectured dualities in these theories. In this paper, we compute the two, three and four-point functions of the operators in the spin zero multiplet of mathcal{N} = 2 Supersymmetric vector matter Chern-Simons theory at large N to all orders of ’t Hooft coupling. While the two- and three-point functions are computed by solving the Schwinger-Dyson equation, this method becomes intractable for the computation of the four-point functions. Thereby, we use bootstrap method to evaluate four-point function of scalar operator {J}_0^f=overline{psi}psi and {J}_0^b=overline{phi}phi . Interestingly, because leftlangle {J}_0^f{J}_0^f{J}_0^brightrangle is a contact term, the four point function of {J}_0^f operator resembles the corresponding correlation function in the free theory, up to overall coupling constant dependent factors and up to some ‘bulk AdS’ contact terms. On the other hand the {J}_0^b four-point function receives an additional contribution compared to the free theory expression due to the {J}_0^f exchange. We find that the double discontinuity of this single trace operator {J}_0^f vanishes and hence it only contributes to AdS-contact term.

Three-point functions at strong coupling in the BMN limit

We consider structure constants of single-trace operators at strong coupling in planar mathcal{N} = 4 SYM theory using the hexagon formalism. We concentrate on heavy-heavy- light correlators where the heavy operators are BMN operators, with large R-charges and finite anomalous dimensions, and the light one is a finite-charge chiral primary operator. They describe the couplings between two highly boosted strings and a supergravity mode in the bulk dual. In the hexagon framework, two sums over virtual magnons are needed to bind the hexagons together around the light operator. We evaluate these sums explicitly at strong coupling, for a certain choice of BMN operators, and show that they factorise into a ratio of Gamma functions and a simple stringy prefactor. The former originates from giant mirror magnons scanning the AdS geometry while the latter stems from small fluctuations around the BMN vacuum. The resulting structure constants have poles at positions where an enhanced mixing with double-trace operators is expected and zeros whenever the process is forbidden by supersymmetry. We also discuss the transition to the classical regime, when the length of the light operator scales like the string tension, where we observe similitudes with the Neumann coefficients of the pp-wave String Field Theory vertex.

The fate of the Konishi multiplet in the \u03b2-deformed Quantum Spectral Curve

We investigate the solution space of the β-deformed Quantum Spectral Curve by studying a sample of solutions corresponding to single-trace operators that in the undeformed theory belong to the Konishi multiplet. We discuss how to set the precise boundary conditions for the leading Q-system for a given state, how to solve it, and how to build perturbative corrections to the Pμ-system. We confirm and add several loop orders to known results in the literature.

Exact Three-Point Functions of Determinant Operators in Planar N=4 Supersymmetric Yang-Mills Theory.

We introduce a nonperturbative approach to correlation functions of two determinant operators and one nonprotected single-trace operator in planar N=4 supersymmetric Yang-Mills theory. Based on the gauge-string duality, we propose that they correspond to overlaps on the string world sheet between an integrable boundary state and a state dual to the single-trace operator. We determine the boundary state using the symmetry and integrability of the dual superstring σ model and write expressions for the correlators at finite coupling, which we conjecture to be valid for operators of arbitrary size. The proposal is put to the test at weak coupling.

Absorption of closed strings by giant gravitons

A new approach to the computation of correlation functions involving two determinant operators as well as one non-protected single trace operator has recently been developed by Jiang, Komatsu and Vescovi. This correlation function provides the holographic description of the absorption of a closed string by a giant graviton. The analysis has a natural interpretation in the framework of group representation theory, which admits a generalization to general Schur polynomials and restricted Schur polynomials. This generalizes the holographic description to any giant or dual giant gravitons which carry more than one angular momentum on the sphere. For a restricted Schur polynomial labeled by a column with N boxes (dual to a maximal giant graviton) we find evidence in favor of integrability. Since this restricted Schur polynomial is a frac{1}{2} -BPS operator, this integrability is a corollary of the results of Jiang, Komatsu and Vescovi.

Holographic correlators in AdS3 without Witten diagrams

We present a formula for the holographic 4-point correlators in AdS3 × S3 involving four single-trace operators of dimension k, k, l, l. As an input we use the super-gravity results for the Heavy-Heavy-Light-Light correlators that can be derived by studying the linear fluctuations around known asymptotically AdS3 × S3 geometries. When the operators of dimension k and l are in the same multiplet there are contributions due to the exchange of single-trace operators in the t and u-channels, which are not captured by the approach mentioned above. However by rewriting the s-channel results in Mellin space we obtain a compact expression for the s-channel contribution that makes it possible to conjecture a formula for the complete result. We discuss some consistency checks that our proposal meets.

AdS3 holography at dimension two

Holography can provide a microscopic interpretation of a gravitational solution as corresponding to a particular CFT state: the asymptotic expansion in gravity encodes the expectation values of operators in the dual CFT state. Such a correspondence is particularly valuable in black hole physics. We study supersymmetric D1-D5-P black holes, for which recently constructed microstate solutions known as “superstrata” provide strong motivation to derive the explicit D1-D5 holographic dictionary for CFT operators of total dimension two. In this work we derive the explicit map between one-point functions of scalar chiral primaries of dimension (1, 1) and the asymptotic expansions of families of asymptotically AdS3 ×S3 × ℳ supergravity solutions, with ℳ either T4 or K3. We include all possible mixings between single-trace and multi-trace operators. We perform several tests of the holographic map, including new precision holographic tests of superstrata, that provide strong supporting evidence for the proposed dual CFT states.

Light-front bootstrap for Chern-Simons matter theories

We propose a new approach to solve conformal field theories and apply it to Chern-Simons Matter theories and three-dimensional bosonization duality. All three-point correlation functions of single-trace operators are obtained in the large-N as a simple application. The idea is to construct, as an effective weakly-coupled theory, a nonlinear realization of the conformal algebra in terms of physical, gauge-invariant, operators. The efficiency of the method is also in the use of an analog of the light-cone gauge and of the momentum-space on the CFT side. AdS/CFT is used as a convenient regulator and as a source of the canonical bracket. The uniqueness of the nonlinear realization manifests the three-dimensional bosonization duality at this order. We also find two more non-unitary solutions which should be analogous to the fishnet theories. The results can also be viewed as an explicit realization of the slightly-broken higher spin symmetry.As a by-product, the cubic action of the Higher Spin Gravity in AdS4 is constructed. While generic Higher Spin Gravities are obstructed at higher orders by nonlocality, we point out the existence of two especially simple and well-defined theories: chiral and anti-chiral whose three-point functions correspond to the two new solutions. These two theories are supposed to give a quantum complete and local example of gravitational bulk duals.

AdS3 four-point functions from frac{1}{8} -BPS states

We compute four-point functions in the Heavy-Heavy-Light-Light limit involving a large family of frac{1}{8} -BPS heavy states whose dual supergravity solutions are explicitly known, avoiding the use of Witten diagrams. This is achieved by using the AdS/CFT dictionary of type IIB supergravity on AdS3 × S3 × ℳ4 that maps supersymmetric heavy operators whose conformal dimension is the order of the central charge to explicit asymp-totically AdS supergravity solutions. Using the Ward Identities for the generators of the mathcal{N}=left(4,4right) superconformalSU(2)Kac-Moodyalgebra,wecanrelateallofthesefour-point functions to each other and to other known four-point functions involving frac{1}{4} -BPS heavy states, furnishing non-trivial checks of the computations. Finally, the Ward Identities can be employed to reconstruct the all-light four-point functions, providing the first holographic correlators of single-trace operators computed in AdS3 involving frac{1}{8} -BPS operators.

Resurgence of one-point functions in a matrix model for 2D type IIA superstrings

In the previous papers, the authors pointed out correspondence between a supersymmetric double-well matrix model and two-dimensional type IIA superstring theory on a Ramond-Ramond background. This was confirmed by agreement between planar correlation functions in the matrix model and tree-level amplitudes in the superstring theory. Furthermore, in the matrix model we computed one-point functions of single-trace operators to all orders of genus expansion in its double scaling limit, and found that the large-order behavior of this expansion is stringy and not Borel summable. In this paper, we discuss resurgence structure of these one-point functions and see cancellations of ambiguities in their trans-series. More precisely, we compute both series of ambiguities arising in a zero-instanton sector and in a one-instanton sector, and confirm how they cancel each other. In case that the original integration contour is a finite interval not passing through a saddle point, we have to choose an appropriate integration path in order for resurgence to work.

Holographic correlators in AdS3

We derive the four-point correlators of scalar operators of dimension one in the supergravity limit of the D1D5 CFT holographically dual to string theory on AdS3×S3×ℳ, with ℳ either T4 or K3. We avoid the use of Witten diagrams but deduce our result from a limit of the heavy-heavy-light-light correlators computed in [1], together with several consistency requirements of the OPE in the various channels. This result represents the first holographic correlators of single-trace operators computed in AdS3.