Chiral Primary Operators Research Articles

Four-point correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM from AdS5 bubbling geometries

Four-point correlation functions are observables of significant interest in holographic field theories. We compute an infinite family of four-point correlation functions of operators in short multiplets of 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory in the supergravity regime, by studying the quadratic fluctuations around non-trivial supergravity backgrounds. The supergravity backgrounds are supersymmetric smooth geometries in the family derived by Lin, Lunin and Maldacena. The light probes comprise an infinite sequence of Kaluza-Klein harmonics of the dilaton/axion. For generic parameter values, the supergravity backgrounds are dual to heavy CFT states. However we focus on the limit in which the dual CFT states become light single-particle states. The resulting all-light four-point correlators are related by superconformal Ward identities to previously known four-point correlators of half-BPS chiral primary operators. By verifying that the Ward identities are satisfied, we confirm the validity of the supergravity method.

The geometry of large charge multi-traces in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM

We construct a one-parameter family of half-BPS solutions of type IIB supergravity using a consistent truncation to gauged five-dimensional supergravity. For small values of the parameter, the solution reduces to the linear perturbation of AdS5 × S5 dual to the chiral primary operator in the stress-tensor multiplet, and we give evidence that the geometry is regular and asymptotes AdS in a normalisable way for arbitrarily large values of the parameter. We conjecture that the solution is the gravity dual of a “heavy” multi-trace operator in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SU(N) Super Yang-Mills made by p copies of the stress-tensor chiral primary operator, with p of order N2 in the large N limit. We perform some holographic checks supporting this duality map.

Giant correlators at quantum level

We compute four-point functions with two maximal giant gravitons and two chiral primary operators at three-loop order in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super-Yang-Mills theory. The Lagrangian insertion method, together with symmetries of the theory fix the integrand up to a few constants. Under the additional assumption of planarity, all these constants can be determined by lower loop data. The obtained result can be regarded as the planar diagram contribution to the full four-point function and is written in terms of the known three-loop conformal integrals. From the four-point function, we extract the OPE coefficients of two giant gravitons and one non-BPS twist-2 operator with arbitrary spin at three-loops, given in terms of harmonic sums. We observe an intriguingly simple relation between the giant graviton OPE coefficients and the OPE coefficients of three single-trace operators.

Superconformal two-point functions of the Nahm pole defect in [formula omitted] super-Yang-Mills theory

We derive a manifestly superconformal expression for the leading-order two-point functions of all single trace chiral primary operators in 4d N=4 super-Yang-Mills theory with a co-dimension one Nahm pole defect. Notably, our result involves only a finite number of superblocks that correspond to 1/2-BPS representations in the defect channel. The derivation builds on a non-trivial exact result for traces of fuzzy spherical harmonics in combination with powers of su(2) generators, which leads to a remarkably compact expression for the bulk-to-defect couplings. Apart from being interesting in its own right as a closed-form expression at finite N, the final result is an essential prerequisite for applying the defect superconformal bootstrap programme to this model.

Equivariant localization for AdS/CFT

We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline-Vergne-Atiyah-Bott fixed point formula. We illustrate the formalism by considering AdS5 × M6 and AdS3 × M8 solutions of D = 11 supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.

On the spectrum and structure constants of short operators in N=4 SYM at strong coupling

We study short operators in planar mathcal{N} = 4 SYM at strong coupling, for general spin and SO(6) symmetric traceless representations. At strong coupling their dimension grows like Delta sim 2sqrt{delta }{lambda}^{1/4} and their spectrum of degeneracies can be analysed by considering the massive spectrum of type II strings in flat space-time. We furthermore compute their structure constants with two arbitrary chiral primary operators. This is done by considering the four-point correlator of arbitrary chiral primary operators at strong coupling in planar mathcal{N} = 4 SYM, including the supergravity approximation plus the infinite tower of stringy corrections that contributes in the flat space limit. Our results are valid for generic rank n symmetric traceless representations of SO(6) and in particular for n ≫ 1, as long as n ≪ λ1/4.

Conformal dispersion relations for defects and boundaries

We derive a dispersion relation for two-point correlation functions in defect conformal field theories. The correlator is expressed as an integral over a (single) discontinuity that is controlled by the bulk channel operator product expansion (OPE). This very simple relation is particularly useful in perturbative settings where the discontinuity is determined by a subset of bulk operators. In particular, we apply it to holographic correlators of two chiral primary operators in \mathcal{N}=4𝒩=4 Super Yang-Mills theory in the presence of a supersymmetric Wilson line. With a very simple computation, we are able to reproduce and extend existing results. We also propose a second relation, which reconstructs the correlator from a double discontinuity, and is controlled by the defect channel OPE. Finally, for the case of codimension-one defects (boundaries and interfaces) we derive a dispersion relation which receives contributions from both OPE channels and we apply it to the boundary correlator in the O(N) critical model. We reproduce the order \epsilon^2ϵ2 result in the \epsilonϵ-expansion using as input a finite number of boundary CFT data.

Holographic operator product expansion of loop operators in the super Yang-Mills theory* *Supported in part by the National Natural Science Foundation of China (11975164, 11935009) and Natural Science Foundation of Tianjin (20JCYBJC00910, 20JCQNJC02030)

In this study, we compute the correlation functions of Wilson(-'t Hooft) loops with chiral primary operators in the supersymmetric Yang-Mills theory with gauge symmetry, which has a holographic dual description of the Type IIB superstring theory on the background. Specifically, we compute the coefficients of the chiral primary operators in the operator product expansion of Wilson loops in the fundamental representation, Wilson-'t Hooft loops in the symmetric representation, Wilson loops in the anti-fundamental representation, and Wilson loops in the spinor representation. We also compare these results to those of the super Yang-Mills theory.

A derivation of the planar limit of mathcal{N} = 2 chiral correlators

We derive analytically the terms of maximal transcendentality of the planar 2- and 3-point functions of single-trace chiral primary operators of mathcal{N} = 2 SQCD on ℝ4, to all orders in the ’t Hooft coupling. These results prove two conjectures we formulated in previous work. Furthermore, we also provide an explicit expression for the terms in the planar 2-point functions of these operators that contain products of two values of the ζ function.

Wilson loops in large symmetric representations through a double-scaling limit

We derive exact formulas for circular Wilson loops in the mathcal{N} = 4 and mathcal{N} = 2* theories with gauge groups U(N) and SU(N) in the k-fold symmetrized product representation. The formulas apply in the limit of large k and small Yang-Mills coupling g, with fixed effective coupling κ ≡ g2k, and for any finite N. In the SU(2) and U(2) cases, closed analytic formulas are obtained for any k, while the 1/k series expansions are asymptotic. In the N ≫ 1 limit, with N ≪ k, there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the k-symmetric Wilson loops and chiral primary operators are also given.

The planar limit of mathcal{N} = 2 chiral correlators

We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of mathcal{N} = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

Notes on gravity multiplet correlators in AdS3 \xd7 S3

We present a compact formula in Mellin space for the four-point tree-level holographic correlators of chiral primary operators of arbitrary conformal weights in (2, 0) supergravity on AdS3× S3, with two operators in tensor multiplet and the other two in gravity multiplet. This is achieved by solving the recursion relation arising from a hidden six-dimensional conformal symmetry. We note the compact expression is obtained after carefully analysing the analytic structures of the correlators. Various limits of the correlators are studied, including the maximally R-symmetry violating limit and flat-space limit.

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.

On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators

Starting with some known localization (matrix model) representations for correlators involving 1/2 BPS circular Wilson loop mathcal{W} in mathcal{N} = 4 SYM theory we work out their 1/N expansions in the limit of large ’t Hooft coupling λ. Motivated by a possibility of eventual matching to higher genus corrections in dual string theory we follow arXiv:2007.08512 and express the result in terms of the string coupling {g}_{mathrm{s}}sim {g}_{mathrm{YM}}^2sim lambda /N and string tension Tsim sqrt{lambda } . Keeping only the leading in 1/T term at each order in gs we observe that while the expansion of leftlangle mathcal{W}rightrangle is a series in {g}_{mathrm{s}}^2/T , the correlator of the Wilson loop with chiral primary operators {mathcal{O}}_J has expansion in powers of {g}_{mathrm{s}}^2/{T}^2 . Like in the case of leftlangle mathcal{W}rightrangle where these leading terms are known to resum into an exponential of a “one-handle” contribution sim {g}_{mathrm{s}}^2/T , the leading strong coupling terms in leftlangle {mathcal{WO}}_Jrightrangle sum up to a simple square root function of {g}_{mathrm{s}}^2/{T}^2 . Analogous expansions in powers of {g}_{mathrm{s}}^2/T are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

On three-point functions in ABJM and the latitude Wilson loop

I consider three-point functions of twist-one operators in ABJM at weak coupling. I compute the structure constant of correlators involving one twist-one un-protected operator and two protected ones for a few finite values of the spin, up to two-loop order. As an application I enforce a limit on the gauge group ranks, in which I relate the structure constant for three chiral primary operators to the expectation value of a supersymmetric Wilson loop. Such a relation is then used to perform a successful five-loop test on the matrix model conjectured to describe the supersymmetric Wilson loop.

The CFT_6 origin of all tree-level 4-point correlators in AdS_3 times S^3

We provide strong evidence that all tree-level 4-point holographic correlators in hbox {AdS}_3 times S^3 are constrained by a hidden 6D conformal symmetry. This property has been discovered in the hbox {AdS}_5 times S^5 context and noticed in the tensor multiplet subsector of the AdS_3 times S^3 theory. Here we extend it to general AdS_3 times S^3 correlators which contain also the chiral primary operators of spin zero and one that sit in the gravity multiplet. The key observation is that the 6D conformal primary field associated with these operators is not a scalar but a self-dual 3-form primary. As an example, we focus on the correlators involving two fields in the tensor multiplets and two in the gravity multiplet and show that all such correlators are encoded in a conformal 6D correlator between two scalars and two self-dual 3-forms, which is determined by three functions of the cross ratios. We fix these three functions by comparing with the results of the simplest correlators derived from an explicit supergravity calculation.

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.

Holography of massive M2-brane theory with discrete torsion

We investigate the gauge/gravity duality between the mathcal{N} = 6 mass-deformed ABJ theory with hbox {U}_k(N+l)times hbox {U}_{-k}(N) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1) times SO(4)/{mathbb Z}_ktimes SO(4)/{{mathbb {Z}}}_k isometry and the discrete torsion l. For chiral primary operators with conformal dimensions Delta =1,2, we obtain the exact vacuum expectation values using the holographic method in 11-dimensional supergravity and show that the results depend on the shapes of droplet pictures in LLM geometries. The frac{l}{sqrt{N}} contributions from the discrete torsion l for several simple droplet pictures in the large N limit are determined in holographic vacuum expectation values. We also explore the effects of the orbifolding {{mathbb {Z}}}_k and the asymptotic discrete torsion l, on the gauge/gravity duality dictionary and on the nature of the asymptotic limits of the LLM geometries.

One point functions for black hole microstates

We compute one point functions of chiral primary operators in the D1–D5 orbifold CFT, in classes of states corresponding to microstates of two and three charge black holes. Black hole microstates describable by supergravity solutions correspond to coherent superpositions of states in the orbifold theory and we develop methods for approximating one point functions in such superpositions in the large N limit. We show that microstates built from long strings (large twist operators) have one point functions that are suppressed by powers of N. Accordingly, even when these microstates admit supergravity descriptions, the characteristic scales in these solutions are comparable to higher derivative corrections to supergravity.

Four-point functions of all-different-weight chiral primary operators in the supergravity approximation

Recently a Mellin-space formula was conjectured for the form of correlation functions of 1/2 BPS operators in planar mathcal{N}=4 SYM in the strong ’t Hooft coupling limit. In this work we report on the computation of two previously unknown four-point functions of operators with weights 〈2345〉 and 〈3456〉, from the effective type-IIB supergravity action using AdS/CFT. These correlators are novel: they are the first correlators with all different weights and in particular 〈3456〉 is the first next-next-next-to-extremal correlator to ever have been computed. We also present simplifications of the known algorithm, without which these computations could not have been executed. These simplifications consist of a direct formula for the exchange part and for the contact part of the correlation function, as well as a simplification of the C tensor algorithm to compute a tensors. After bringing our results in the appropriate form we successfully corroborate the recently conjectured formula.