ABJ Theory Research Articles

Holographic 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS surface defects in ABJM

We study the class of AdS3 × ℂℙ3 solutions to massive Type IIA supergravity with osp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{osp} $$\\end{document}(6|2) superconformal algebra recently constructed in [1]. These solutions are foliations over an interval preserving 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} = (0, 6) supersymmetry in two dimensions, that in the massless limit can be mapped to the AdS4 × ℂℙ3 solution of ABJM/ABJ. We show that in the massive case extra NS5-D8 branes, that we interpret as 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS surface defect branes within the ABJ theory, backreact in the geometry and turn one of the 3d field theory directions onto an energy scale, generating a flow towards a 2d CFT. We construct explicit quiver field theories that we propose flow in the IR to the (0, 6) SCFTs dual to the solutions. Finally, we show that the AdS3 solutions realise geometrically, in terms of large gauge transformations, an extension to the massive case of Seiberg duality in ABJ theories proposed in the literature.

Duality between U × U and O × USp theories from A3 = D3

We study the duality between the ABJ(M) theory at Chern-Simons level k = 4 and the orientifold ABJ theory at Chern-Simons level k = 1 by using the S3 partition function. The partition function can be computed using the supersymmetric localization in terms of a matrix model, and we derive an ideal Fermi gas system by applying the Fermi gas formalism to the matrix model. By using this formalism, we show that the matrix model can be embedded in the partition function of Â\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{A} $$\\end{document}3 or D̂\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{D} $$\\end{document}3 quiver theories, and the equality of these quiver theories, which comes from Â\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{A} $$\\end{document}3 = D̂\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{D} $$\\end{document}3, leads to exact relations of the matrix models. These exact relations can be used to check the identification of the flavor symmetries. The perfect agreement between the matrix models also implies that no decoupled sector is needed.

Discrete global symmetries: gauging and twisted compactification

Discrete global symmetries 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} = 2 SCFTs are studied via two operations: gauging and twisted compactification. We consider gauging of discrete symmetries in several well-known 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} = 2 SCFTs, including SU(n) SQCD with 2n flavors, theories of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} of type A2n−1, and Argyres-Douglas theories of type (AN, AN), as well as propose new 4d SCFTs as a result. The wreathing technique, which involves gauging a subgroup of the automorphism group of the quiver diagram of the corresponding 3d mirror theory, is exploited. This allows us to understand several properties of discretely gauged theories, including moduli spaces and how discrete gauging affects the mixed ’t Hooft anomaly between the 1-form symmetry and the 0-form flavor symmetry. Many examples are viewed through the lens of the Argyres-Seiberg duality and its generalization. We also examine discrete gauging of SU(2) SQCD with 4 flavors by various ℤ2 and ℤ2 × ℤ2 subgroups of the permutation group S4 using the superconformal index. Regarding compactification, we propose a magnetic quiver for 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} = 2 SU(n) SQCD with 2n flavors compactified on a circle with a ℤ2 twist. The twisted compactification by non-invertible symmetries of the 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 SYM theory with gauge group SU(N) is revisited. The non-invertible symmetry naturally gives rise to a ℤk action on the scalar fields parametrizing the moduli space. Upon examining the ℤk invariant chiral ring of the Higgs branch, we find that, in addition to the largest branch of the moduli space that is expected to be captured by the ABJ(M) theory, there exist in general nilpotent operators that lead to a branch of the moduli space which is a radical ideal.

Quantized strings and instantons in holography

We study worldsheet instantons in holographic type IIA backgrounds directly in string theory. The first background is a dimensional reduction of AdS7 × S4 and is dual to the maximally supersymmetric Yang-Mills theory on S5. The second background is AdS4 × CP3 dual to ABJM in the type IIA limit. We compute the one-loop partition function of the fundamental string in these backgrounds and show that the result is in exact agreement with field theory predictions. We argue that for higher rank instantons, the string partition function takes a product form of the single instanton partition function times the contribution of two orbifolds on the worldsheet. We determine the orbifold factor to be n−3/2 where n is the instanton rank. With this result, we reproduce the series of non-perturbative corrections in α′ to the planar S5 free energy. When studying the worldsheet instanton partition function on CP3, we encounter twelve fermionic and twelve bosonic zero modes. By deforming the ABJM theory, the zero-modes are lifted and consequently the tower of worldsheet instantons can be evaluated and matched to known results in the QFT. As a by-product, we determine a series of higher rank instanton corrections to the free energy of the mass-deformed and orbifolded ABJ(M) theory.

Interpolating Wilson loops and enriched RG flows

We study new 1/24 BPS circular Wilson loops in ABJ(M) theory, which are defined in terms of several parameters that continuously interpolate between previously known 1/6 BPS loops (both bosonic and fermionic) and 1/2 BPS fermionic loops. We compute the expectation value of these operators up to second order in perturbation theory using a one-dimensional effective field theory approach. Within dimensional regularization, we find non-trivial β-functions for the parameters, which are marginally relevant deformations triggering RG flows from a UV fixed point represented by the 1/6 BPS bosonic loop to an IR fixed point represented by a 1/2 BPS fermionic loop. Generically, along all flows at least one supercharge of the theory is preserved, so that we refer to them as enriched RG flows. In particular, fixed points are connected through 1/6 BPS fermionic operators. This holds at framing zero, which is a consequence of the regularization scheme employed. We also establish the validity of the g-theorem, relating the expectation values of the Wilson loops corresponding to the UV and IR fixed points of the flow, and discuss the one-dimensional defect SCFT living on the Wilson loop contour.

Constant primary operators and where to find them: the strange case of BPS defects in ABJ(M) theory

We investigate the one-dimensional defect SCFT defined on the 1/2 BPS Wilson line/loop in ABJ(M) theory. We show that the supermatrix structure of the defect imposes a covariant supermatrix representation of the supercharges. Exploiting this covariant formulation, we prove the existence of a long multiplet whose highest weight state is a constant supermatrix operator. At weak coupling, we study this operator in perturbation theory and confirm that it acquires a non-trivial anomalous dimension. At strong coupling, we conjecture that this operator is dual to the lowest bound state of fluctuations of the fundamental open string in AdS4 × ℂℙ3 around the classical 1/2 BPS solution. Quite unexpectedly, this operator also arises in the cohomological equivalence between bosonic and fermionic Wilson loops. We also discuss some regularization subtleties arising in perturbative calculations on the infinite Wilson line.

On refined Chern\u2013Simons and refined ABJ matrix models

We consider the matrix model of U(N) refined Chern–Simons theory on S^3 for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality qleftrightarrow t^{-1} which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

ABJM at strong coupling from M-theory, localization, and Lorentzian inversion

We study the stress tensor multiplet four-point function in the 3d maximally supersymmetric ABJ(M) theory with Chern-Simons level k = 2, which in the large N limit is holographically dual to weakly coupled M-theory on AdS4 × S7/ℤ2. We use the Lorentzian inversion to compute the 1-loop correction to this holographic correlator coming from Witten diagrams with supergravity R and the first higher derivative correction R4 vertices, up to a finite number of contact terms that contribute to low spins where the inversion formula does not converge. We find a precise match with the corresponding terms in the 11d M-theory S-matrix by taking the flat space limit, which is not sensitive to these contact terms. We then conjecturally fix these contact terms by analytically continuing the inversion formula below its expected range of convergence, and verify this conjecture using supersymmetric localization. Finally, we compare some of the 1-loop CFT data to non-perturbative in N bounds from the numerical conformal bootstrap, which we compute at unprecedently high accuracy, and find that the 1-loop corrections saturate the bounds in the large N regime, which extends the previously observed match at tree level.

Squashing and supersymmetry enhancement in three dimensions

We consider mass-deformed theories with \mathcal{N}\geq2𝒩≥2 supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the 3d partition function can be obtained by a limit of a 4d index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently observed squashing independence of the partition function of mass-deformed ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various 3d supersymmetric theories.

Notes on hyperloops in mathcal{N} = 4 Chern-Simons-matter theories

We present new circular Wilson loops in three-dimensional mathcal{N} = 4 quiver Chern-Simons-matter theory on S3. At any given node of the quiver, a two-parameter family of operators can be obtained by opportunely deforming the 1/4 BPS Gaiotto-Yin loop. Including then adjacent nodes, the coupling to the bifundamental matter fields allows to enlarge this family and to construct loop operators based on superconnections. We discuss their classification, which depends on both discrete data and continuous parameters subject to an identification. The resulting moduli spaces are conical manifolds, similar to the conifold of the 1/6 BPS loops of the ABJ(M) theory.

The topological line of ABJ(M) theory

We construct the one-dimensional topological sector of mathcal{N} = 6 ABJ(M) theory and study its relation with the mass-deformed partition function on S3. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at two-loop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge cT of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs.

ABJ correlators with weakly broken higher spin symmetry

We consider four-point functions of operators in the stress tensor multiplet of the 3d mathcal{N} = 6 U(N)k× U(N + M)−k or SO(2)2k× USp(2 + 2M)−k ABJ theories in the limit where M and k are taken to infinity while N and λ ∼ M/k are held fixed. In this limit, these theories have weakly broken higher spin symmetry and are holographically dual to mathcal{N} = 6 higher spin gravity on AdS4, where λ is dual to the bulk parity breaking parameter. We use the weakly broken higher spin Ward identities, superconformal Ward identities, and the Lorentzian inversion formula to fully determine the tree level stress tensor multiplet four-point function up to two free parameters. We then use supersymmetric localization to fix both parameters for the ABJ theories in terms of λ, so that our result for the tree level correlator interpolates between the free theory at λ = 0 and a parity invariant interacting theory at λ = 1/2. We compare the CFT data extracted from this correlator to a recent numerical bootstrap conjecture for the exact spectrum of U(1)2M× U(1 + M)−2M ABJ theory (i.e. λ = 1/2 and N = 1), and find good agreement in the higher spin regime.

Squashing, mass, and holography for 3d sphere free energy

We consider the sphere free energy F(b; mI) in mathcal{N} = 6 ABJ(M) theory deformed by both three real masses mI and the squashing parameter b, which has been computed in terms of an N dimensional matrix model integral using supersymmetric localization. We show that setting {m}_3=ifrac{b-{b}^{-1}}{2} relates F(b; mI) to the round sphere free energy, which implies infinite relations between mI and b derivatives of F(b; mI) evaluated at mI = 0 and b = 1. For mathcal{N} = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of m1, m2, which were previously computed to all orders in 1/N using the Fermi gas method. This allows us to compute {partial}_b^4Fleft|{}_{b=1}right. and {partial}_b^5Fleft|{}_{b=1}right. to all orders in 1/N, which we precisely match to a recent prediction to sub-leading order in 1/N from the holographically dual AdS4 bulk theory.

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.

The M-theory archipelago

We combine supersymmetric localization results and the numerical conformal bootstrap technique to study the 3d maximally supersymmetric ( mathcal{N} = 8) CFT on N coincident M2-branes (the U(N)k × U(N)−k ABJM theory at Chern-Simons level k = 1). In particular, we perform a mixed correlator bootstrap study of the superconformal primaries of the stress tensor multiplet and of the next possible lowest-dimension half-BPS multiplet that is allowed by 3d mathcal{N} = 8 superconformal symmetry. Of all known 3d mathcal{N} = 8 SCFTs, the k = 1 ABJM theory is the only one that contains both types of multiplets in its operator spectrum. By imposing the values of the short OPE coefficients that can be computed exactly using supersymmetric localization, we are able to derive precise islands in the space of semi-short OPE coefficients for an infinite number of such coefficients. We find that these islands decrease in size with increasing N. More generally, we also analyze 3d mathcal{N} = 8 SCFT that contain both aforementioned multiplets in their operator spectra without inputing any additional information that is specific to ABJM theory. For such theories, we compute upper and lower bounds on the semi-short OPE coefficients as well as upper bounds on the scaling dimension of the lowest unprotected scalar operator. These latter bounds are more constraining than the analogous bounds previously derived from a single correlator bootstrap of the stress tensor multiplet. This leads us to conjecture that the U(N)2 × U(N + 1)−2 ABJ theory, and not the k = 1 ABJM theory, saturates the single correlator bounds.

Structure constants of heavy operators in ABJM and ABJ theory

Efficient and powerful approaches to the computation of correlation functions involving determinant, sub-determinant and permanent operators, as well as traces, have recently been developed in the setting of ${\cal N}=4$ super Yang-Mills theory. In this article we show that they can be extended to ABJM and ABJ theory. After making use of a novel identity which follows from character orthogonality, an integral representation of certain projection operators used to define Schur polynomials is given. This integral representation provides an effective description of the correlation functions of interest. The resulting effective descriptions have ${1\over N}$ as the loop counting parameter, strongly suggesting their relevance for holography.

A matrix model for the latitude Wilson loop in ABJM theory

In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS Wilson loops on a latitude circular contour, so providing a new weak-strong interpolation tool. Intriguingly, the matrix model turns out to be a particular case of that computing torus knot invariants in U(N1|N2) Chern-Simons theory. At weak coupling we check our proposal against a three-loop computation, performed for generic framing, winding number and representation. The matrix model is amenable of a Fermi gas formulation, which we use to systematically compute the strong coupling and genus expansions. For the fermionic Wilson loop the leading planar behavior agrees with a previous string theory prediction. For the bosonic operator our result provides a clue for finding the corresponding string dual configuration. Our matrix model is consistent with recent proposals for computing Bremsstrahlung functions exactly in terms of latitude Wilson loops. As a by-product, we extend the conjecture for the exact B1/6θ Bremsstrahlung function to generic representations and test it with a four-loop perturbative computation. Finally, we propose an exact prediction for B1/2 at unequal gauge group ranks.

AdS4/CFT3 for unprotected operators

We consider the four-point function of the lowest scalar in the stress-energy tensor multiplet in mathcal{N}=8 ABJ(M) theory [1, 2]. At large central charge cT ∼ N3/2, this correlator is given by the corresponding holographic correlation function in 11d supergravity on AdS4 × S7. We use Mellin space techniques to compute the leading 1/cT correction to anomalous dimensions and OPE coefficients of operators that appear in this holographic correlator. For half and quarter-BPS operators, we find exact agreement with previously computed localization results. For the other BPS and non-BPS operators, our results match the mathcal{N}=8 numerical bootstrap for ABJ(M) at large cT, which provides a precise check of unprotected observables in AdS/CFT.

WiLE: A Mathematica package for weak coupling expansion of Wilson loops in ABJ(M) theory

We present WiLE, a Mathematica® package designed to perform the weak coupling expansion of any Wilson loop in ABJ(M) theory at arbitrary perturbative order. For a given set of fields on the loop and internal vertices, the package displays all the possible Feynman diagrams and their integral representations. The user can also choose to exclude non planar diagrams, tadpoles and self-energies. Through the use of interactive input windows, the package should be easily accessible to users with little or no previous experience. The package manual provides some pedagogical examples and the computation of all ladder diagrams at three-loop relevant for the cusp anomalous dimension in ABJ(M). The latter application gives also support to some recent results computed in different contexts. Program summaryProgram Title: WiLEProgram Files doi:http://dx.doi.org/10.17632/xm53d23hph.1Licensing provisions: GNU LGPLProgramming language: Wolfram Mathematica version 8.0 or higherNature of problem: In ABJ(M) theory one can define bosonic and fermionic Wilson loops. In order to study those operators using the standard perturbation theory techniques, one has to expand the Wilson loops at weak coupling and perform all the Wick contractions. Since the theory possesses a quiver structure and its Lagrangian contains many different vertices, this procedure could be very involved. Then we propose a Mathematica package to automatize this computation.Solution method: The aim of the package WiLE is to obtain the weak coupling expansion of the Wilson loops in terms of Feynman diagrams. Given a set of initial data, as the set of vertices involved in the diagram and the fields contributing in the Wilson loop expansion, the package computes the related diagrams and their integral representations.Additional comments: Web page: https://github.com/miciosca/WiLE, contact: michelangelo.preti@desy.de — for bugs, possible improvements and questions.

ABJ quadrality

We study physical consequences of adding orientifolds to the ABJ triality, which is among 3d mathcal{N}=6 superconformal Chern-Simons theory known as ABJ theory, type IIA string in AdS4 × ℂℙ3 and mathcal{N}=6 supersymmetric (SUSY) Vasiliev higher spin theory in AdS4. After adding the orientifolds, it is known that the gauge group of the ABJ theory becomes O(N1) × USp(2N2) while the background of the string theory is replaced by AdS4 × ℂℙ3/Z2, and the supersymmetries in the both theories reduce to mathcal{N}=5 . We propose that adding the orientifolds to the mathcal{N}=6 Vasiliev theory leads to mathcal{N}=5 SUSY Vasiliev theory. It turns out that the mathcal{N}=5 case is more involved because there are two formulations of the mathcal{N}=5 Vasiliev theory with either O or USp internal symmetry. We show that the two mathcal{N}=5 Vasiliev theories can be understood as certain projections of the mathcal{N}=6 Vasiliev theory, which we identify with the orientifold projections in the Vasiliev theory. We conjecture that the O(N1) × USp(2N2) ABJ theory has the two vector model like limits: N2 ≫ N1 and N1 ≫ N2 which correspond to the semi-classical mathcal{N}=5 Vasiliev theories with O(N1) and USp(2N2) internal symmetries respectively. These correspondences together with the standard AdS/CFT correspondence comprise the ABJ quadrality among the mathcal{N}=5 ABJ theory, string/M-theory and two mathcal{N}=5 Vasliev theories. We provide a precise holographic dictionary for the correspondences by comparing correlation functions of stress tensor and flavor currents. Our conjecture is supported by various evidence such as agreements of the spectra, one-loop free energies and SUSY enhancement on the both sides. We also predict the leading free energy of the mathcal{N}=5 Vasiliev theory from the CFT side. As a byproduct, we give a derivation of the relation between the parity violating phase in the mathcal{N}=6 Vasiliev theory and the parameters in the mathcal{N}=6 ABJ theory, which was conjectured in [1].