A Brief Review of Vasiliev Theory on \mathrm{AdS}_{3}

Three-Point Functions in N=2 Higher-Spin Holography

Three-Point Functions in Higher-Spin Holography

Heidar Moradi and Konstantinos Zoubos

moradi@nbi.dk, kzoubos@nbi.dk


Niels Bohr Institute

Blegdamsvej 17, DK-2100 Copenhagen Ø

Denmark


The Kazama-Suzuki models with the non-linear chiral algebra have been conjectured to be dual to the fully supersymmetric Prokushkin-Vasiliev theory of higher-spin gauge fields coupled to two massive multiplets on . We perform a non-trivial check of this duality by computing three-point functions containing one higher-spin gauge field for arbitrary spin and deformation parameter from the bulk theory, and from the boundary using a free ghost system based on the linear algebra. We find an exact match between the two computations. In the ’t Hooft limit, the three-point functions only depend on the wedge subalgebra and the results are equivalent for any theory with such a subalgebra. In the process we also find the emergence of superconformal symmetry near the boundary by computing holographic OPE’s, consistently with a recent analysis of asymptotic symmetries of higher-spin supergravity.


1 Introduction

Since its inception, the AdS/CFT correspondence [1] has been one of the major research directions within the high energy theory community and has evolved into a versatile framework for performing computations within a wide range of strongly coupled systems arising across theoretical physics. However, a proof of the original conjecture is still lacking, hampered by our current lack of understanding of strongly coupled gauge theory on the one hand and of string theory on RR backgrounds on the other. As an intermediate step it is thus desirable to look for simpler versions of the correspondence that exhibit some of its features but bypass many of the complexities of gauge and string theory.

Recently the higher-spin theories of Vasiliev on anti de Sitter space [2, 3] have received a lot of attention. These highly non-linear theories consist of a tower of interacting massless higher-spin fields and are somewhere between conventional field theories and string theory in terms of complexity. They are believed to be related to the tensionless limit of superstring theory (see [4] for some recent developments).

In [5] it was conjectured that Vasiliev’s minimal bosonic theory on with suitable boundary conditions on the bulk scalar field is dual to the free theory of massless scalars in its -singlet sector in the large limit. This was extended by Klebanov and Polyakov [6] to the critical vector model. Recent calculations, such as three-point functions [7], have provided non-trivial evidence for this conjecture. See [8] for a recent review.

An even simpler duality was recently proposed by Gaberdiel and Gopakumar [9]. Motivated by the conjecture of Klebanov and Polyakov, together with the observation [10, 11, 12] that the asymptotic symmetries of the higher-spin generalization of gravity on lead to algebras on the boundary, they proposed an exact duality between minimal models realized as the WZW coset

(1.1)

and the bosonic truncation of Vasiliev higher-spin theory on in the ’t Hooft limit

(1.2)

This duality is simple both due to the usual power of conformal symmetry in two dimensions, which is only enhanced because of the higher-spin symmetry, and because tensor fields with spin greater than one on do not have any bulk degrees of freedom.

Since the original proposal, a large number of tests have been carried out, regarding partition functions [13], higher-spin black hole backgrounds [14, 15, 16, 17], and correlation functions [18, 19, 20, 21]. These investigations have led to slight refinements of the original conjecture and a better understanding of the matching of states between the bulk and boundary theory [22, 23, 24, 25, 26]. Recent reviews of the conjecture and the above developments can be found in [27, 28].

There have also been extensions of the conjecture beyond the original class of minimal model CFTs [29, 30], and proposals with [31] and more recently supersymmetry [32]. The proposal of [31], which we will review below, has already been subjected to several precise checks, such as the large- matching of partition functions [33], detailed analysis of the symmetries [34, 35, 36], and more recently an analysis of the symmetries at the quantum level [37] which revealed an interesting duality structure of equivalent theories at different values of the parameters. For related work based on supergravity, see [38, 39].

In order to put the proposal of [31] on even firmer ground, it is important to move beyond the symmetries and spectrum and compare correlation functions on both sides of the duality. This will be our goal in this paper. In particular, we compute three-point functions holographically, using the higher-spin theory on , as well as directly from the boundary CFT with symmetry. Our three-point functions will be of a restricted type, involving two bosonic matter fields and one bosonic higher-spin field. Despite the restriction to bosonic fields, the richer structure of the theory allows us to compute several types of correlation functions not present in the non-supersymmetric theory. As an illustration of our results, let us display the following three-point function containing massive scalars and a bosonic higher-spin current not present in the non-supersymmetric theory:

For this class of correlation functions, we find precise matching between the bulk and boundary calculations, thus lending further support to the version of the minimal-model/higher-spin correspondence.

The plan of this paper is as follows. In the following section we will review the proposal of [31], motivate a small modification of Vasiliev theory and calculate the scalar masses in this formalism. This will at the same time fix our notation and conventions. In section 3 we will establish the precise AdS/CFT dictionary for the higher-spin fields by, using the bulk theory, deriving operator product expansions of conserved currents of the dual boundary CFT. As a side product, this gives a holographic proof of the emergence of near the boundary and is by itself a consistency check of the duality. In section 4, which forms the main technical part of the paper, we will perform the holographic computation of three-point functions from the higher-spin theory. The corresponding CFT calculation is performed in section 5, where (as already mentioned) precise agreement is found. We conclude with a discussion of future directions and open problems.

Furthermore, we have included two appendices. In appendix A, we will give a lightning review of the full non-linear Prokushkin-Vasiliev theory and its linearization which is used in this paper. Finally, appendix B contains explicit formulas for the structure constants of and algebras, together with certain useful relations and properties used in the paper.

Note Added:

During the completion of this article we became aware of the parallel work [40], which also considers three-point functions in the duality. That work computes three-point functions with fermionic primaries and higher-spin bosonic currents, which are not considered here, and achieves a better understanding of the relation between the bulk and CFT states. Although their bulk approach is similar to ours, our boundary approaches are very different. Furthermore, the holographic OPE’s of section 3 are not considered in [40]. Where there is overlap, we find agreement with the results of [40].

2 The Minimal model – Higher-spin duality

In [31], it was conjectured that the Kazama-Suzuki model [41], which can be represented as an ordinary coset [42]

(2.1)

is dual to the supersymmetric Prokushkin-Vasiliev theory [43, 44] on with the parameter identification1 in the ’t Hooft limit

(2.2)

The notation stands for the untwisted affine Lie algebra associated to , at level .2 Vasiliev theory contains two massive 3d hypermultiplets, and , with two complex scalars and two fermions in each, with the masses

(2.3)

and equal masses for the other multiplet. The two multiplets have slightly different couplings to the massless higher-spin fields. The massless sector can be formulated as a Chern-Simons theory3 [44], where the super higher-spin algebra is an infinite-dimensional Lie algebra which can be thought of as an analytical continuation of to non-integer (see appendix B for more details).

The asymptotic symmetries of a Chern-Simons theory together with Brown-Henneaux-type boundary fall-off conditions [10, 45], translate into the classical Drinfeld-Sokolov reduction [46] of . In the case of pure gravity, , this leads to the Virasoro algebra, while for this leads to the non-linear [35] algebra. On the other hand, in [47] the chiral algebra of the coset (2.1) was shown to be related to quantum Drinfeld-Sokolov reduction of with the principal embedding of , which is the super -algebra . Recently it was shown that in the ’t Hooft limit, the chiral algebra has the limit [37], which is crucial for the duality to hold and for the calculations in this paper.

The restriction of the range of the parameter leads to scalar masses with . It is well known [48] that for this mass range one can choose two different boundary conditions, with the “usual” quantization being the one with the largest value of the conformal dimension. Recall the usual AdS/CFT dictionary between masses and conformal dimensions of dual CFT operators

(2.4)

for massive scalars and spin fermions, respectively. The dual conformal weights are given by [31]

(2.5)

The bosonic operators in the first multiplet correspond to the scalar with the usual quantization and the scalar with the alternative quantization, while the quantizations are opposite in the second multiplet.

Let label the states of the coset (2.1) up to field identifications due to outer automorphisms of the different factors in the coset. Here and are highest weights of and , respectively, while . In the NS sector we have . In [31], it was proposed that the following holomorphic coset primary fields with chiral conformal weights

(2.6)

where is the fundamental representation, can be used to construct the dual fields (2.5) by gluing holomorphic and anti-holomorphic states as follows

(2.7)

and for the other multiplet

(2.8)

In the ’t Hooft limit, the correlation functions we will be considering only depend on the higher-spin algebra . Thus, in section 5.2 we will generate the corresponding highest-weight representations using a free-field CFT having as a subalgebra. Our highest-weight representations will then be constructed in terms of free fields such that they match the above coset primary fields.

2.1 Modified Vasiliev Theory

In this paper we will only consider the linearized Vasiliev theory, in which the matter fields propagate on a fixed higher-spin background on . This means that we will not take into account effects such as backreaction of matter fields on the higher-spin fields and non-linear interactions between the matter fields. See appendix A for a very brief review of how this linearized theory comes out of the full non-linear Vasiliev theory.

The linearized Vasiliev theory is formulated in terms of two spacetime one-forms which can be identified with Chern-Simons gauge fields and describe the tower of higher-spin fields and , and two 0-forms which are generating functions of the matter fields, and . They take values in the associative algebra [49] generated by the spinor and modulo the relations

(2.9)

The product between the so-called deformed oscillators will be denoted by , and accordingly and will denote the commutator and anti-commutator wrt. to the -product, respectively. In this algebra the fields have the following expansions

(2.10)

and similarly for and . The coefficients are symmetrized in the indices and they have Grassmann parity equal to the number of indices mod 2. Thus commutators of elements in the algebra automatically turn into supercommutators of and polynomials. In fact, as we will see in a moment, supercommutators of symmetrized elements of , with the above -grading, form the infinite dimensional Lie superalgebra [44], with the identification .

The generating element is responsible for doubling the number of fields and thereby the extension of the supersymmetry. The invariant subsets are projected out as

(2.11)

The lowest components and correspond to the two complex scalars and two fermions discussed in the previous section, respectively, while the fields with more that two spinor indices form a tower of auxiliary fields. There are four corresponding fields from and all together we have two sets of hypermultiplets

(2.12)

The linearized Vasiliev equations for the matter fields are (see appendix A)

(2.13)

while the equations for the one-forms are just flatness conditions

(2.14)

Note that the flatness conditions only involve (anti-)commutators when written in component form, so if we turn off the matter fields the theory reduces to a Chern-Simons theory which is the higher-spin SUGRA recently studied in [35]. The full associative algebra only enters through coupling to matter fields as seen from the equations (2.13).

It is obvious that this formalism quickly becomes very tedious. We have to multiply pairs of symmetrized elements of , then use the relations (2.9) to express the result as sums of symmetrized products of and . Inspired by the calculation in [19], we would like to have closed-form expressions for the products. This is what we will consider now.

The Lie algebra structure can be inherited from the associative product of an algebra we will call , which can be constructed as the following quotient [50]

(2.15)

where is the universal enveloping algebra of , is its second order Casimir element, and the factor is spanned by the identity element of (see more details in appendix B).4 The Lie algebra contains the following infinite tower of generators,

(2.16)

where while . In addition to these, contains the identity element which we will write as . The generators and form an subalgebra, where is the supercharge. Actually we also get a subalgebra if we add the other supercharge and the generator of R-symmetry . According to (2.15) we can express all the generators (2.16) in terms of the generators, but is actually sufficient since all can be written as anti-commutators of .

It is shown in [50] that can be generated by together with an element (which is essentially the commutator of ) with the properties

(2.17)

where . Using the identifications and

(2.18)

we clearly see that (2.17) and (2.9) are equivalent. Thus is isomorphic to . We can actually directly write down the all the generators (2.16) in terms of and . By looking at the (anti-)commutators of the and together with appendix B of [19], it is clear that the generators are related to the generators by

(2.19)

for integer and

(2.20)

for half-integer . Here is a symmetric product of ’s with of and . We will however not need the explicit mapping between and in this paper, only the fact that they are isomorphic.

In [50] the structure constants of the (linear) -algebra and an associative extension thereof were explicitly constructed. It turns out that is a subalgebra of this. By some work we can extract the structure constants in the form which is convenient for us5, see appendix B for notation and concrete formulas.

We will hereby modify the traditional Vasiliev formalism by changing into . It is convenient to simplify the notation by allowing to be half-integer and identifying

(2.21)

In this formalism the expansions (2.10) of the generating functions are given as

(2.22)

and similarly for and . The notation stands for summation over half-integer steps. Note that we can easily distinguish the bosonic components from the fermionic (anticommuting) ones , since is always an integer while is half of an odd integer. In this formalism the physical scalars and fermions , are given by appropriate superpositions of the lowest components , , and .

In appendix A we mention the fact that, using the projection operator , the bosonic subalgebra of decomposes into , which is isomorphic to . Since the same projector is also used to extract , when computing the bosonic three-point function we only need the subalgebra given by . Therefore, the three-point functions could be extracted from the results of [19] using the relation , where is the parameter appearing in [19]. However, in the fermionic case, we need to use more than just the bosonic subalgebra, and we only know the coefficients in the basis (2.16). In this basis the fermions and bosons do not come out as naturally, so with a view to extending our results to eventually include fermions we choose to perform the full bosonic calculation in this basis.

2.2 Scalars Propagating on

In this section we will illustrate how the Vasiliev equations (2.13) give rise to the Klein-Gordon equation on for the scalars, with the correct masses as known in the literature [43, 31]. In the traditional formalism of Vasiliev based on the deformed oscillators and , the same calculation would be much more tedious.

The connection corresponding to is given as

(2.23)

where we have mapped to the metric formulation by , [10, 45]. The trace is defined and normalized as follows

(2.24)

Turning on other modes, such that (2.14) and appropriate boundary conditions are satisfied, corresponds to higher-spin deformations of . We will for now only consider the scalar fields propagating on , so we will set the fermionic coefficients . Plugging (2.22) into Vasiliev equation (2.13) we find

(2.25)

The coefficients of linearly independent terms should be set to zero individually. Using the properties of the structure constants given in appendix B, we find the following set of coupled equations

(2.26)

Note that we obviously define for modes outside of the wedge . These equations can be solved recursively in order to express the auxiliary fields in terms of and , and thereby find the equations of motion of these scalars. Analyzing the structure of these equations, it turns out that the minimal number of equations needed are

Solving these recursively we can eliminate all the auxiliary fields and reduce to two coupled equations

(2.27)

with the Laplacian of in the coordinates (2.23) given by

(2.28)

In order to bring these equations in standard form, we can remove the term of the second equation by subtracting these two equations with an appropriate weight. This leads to the coupled Klein-Gordon equations

(2.29)

The fields and are clearly not “mass-eigenstates”, but their superpositions must be. Diagonalizing the mass matrix we find

(2.30)

Thus the masses of the two scalars are given by

(2.31)

and from the eigenvectors of the mass matrix we read off the correct superpositions

(2.32)

By rescaling , the masses and , exactly match the results known from the traditional Vasiliev theory [31, 43]. This confirms that our formulation works as expected without the very tedious manipulations involved in the deformed oscillator approach. The advantages of having explicit formulas for the structure constants of cannot be understated: without them our approach, originally laid out in [19], would be very hard to use for extracting three-point functions for arbitrary spin .

With higher-spin deformations of , one can show that the Klein-Gordon equations get higher derivative corrections, as also observed in [19]. We will however not need any of these in this paper.

3 Holographic OPE’s and the AdS/CFT dictionary

Recall that the Brown-Henneaux type asymptotic fall-off conditions [10, 45] translate into classical Drinfeld-Sokolov reduction of the gauge algebra with respect to the embeddings, which correspond to a set of first class constraints. In the so-called lowest-weight gauge [46, 45, 35], the super-connections of constant slices take the form

(3.1)

where only terms of lowest mode are allowed. and are the floor and ceiling operators. Here is the Chern-Simons level6 and the factor is extracted such that the above solution reduces to the BTZ black hole when turning off higher-spin contributions. We choose the normalizations as

(3.2)

The functions and must be holomorphic while and must be anti-holomorphic in order to solve the equations of motion (2.14). In order to calculate correlation functions [52, 53] containing a holomorphic field of spin , we need to add a corresponding source term to the boundary CFT action

(3.3)

Note that the spin field is irrelevant in the renormalization group sense and will therefore change the UV-structure of the dual CFT, which from the bulk perspective corresponds to that the geometry will no longer asymptote to the same geometry.

From the standard prescription of AdS/CFT, the source terms correspond to boundary values of the dual bulk-fields, therefore we need to generalize the boundary conditions. Inspired by the spin-3 case [14], we propose the following generalization of the super-connection

(3.4)

where the functions and are non-chiral functions.7 Following the ideas developed in [14], we will show that evaluating the bulk equations of motion to this ansatz, will yield the Ward identities of the dual CFT in the presence of higher-spin sources. We can in particular show the emergence of symmetry near the boundary, by deriving the OPE’s of the conserved currents in the dual CFT using the bulk theory. This can be thought of as an alternative (and holographic) approach to probing the asymptotic symmetries, which was done in [35] using different means.

The full gauge field is given by

(3.5)

Using the Baker-Campbell-Hausdorff formula

and the fact that is ad-diagonalized in the basis (2.16)

we find the following dependence for each generator in (3.5)

(3.6)

This implies that terms with highest possible modes, and , are the most dominant near the boundary and can thus be regarded as source terms. Note that this is nothing but a Fefferman-Graham expansion of , which happens to be finite. Thus in order to establish the AdS/CFT dictionary, we need to investigate if these terms can be identified with the sources in the boundary action (3.3). It turns out they actually can be identified with the boundary sources (3.3), up to a factor of .

3.1 Flatness Conditions

Using the ansatz (3.4) and the equations of motion, we can collect all the terms into coefficients of the Lie algebra generators

(3.7)

giving rise to the following two set of equations

(3.8)

The coefficients for the bosonic generators are found to be

(3.9)

and for the fermionic generators we have

(3.10)

Here we have used the relations given in equation (B.5) and the step function defined as

(3.11)

Note that the “hat” means we are using the structure constants of , see appendix B for more details. Looking at the form of the equations given by and one can see that by starting from the highest modes, and , we can recursively solve and in terms of the highest modes and , respectively. Finally at the lowest modes, and , the equations of motion are reduced to relations containing only , , and . These equations are the holographic Ward identities in the presence of sources [14, 54, 55], and from these we can identify the correct normalization for the sources by holographically deriving the corresponding OPE’s of the dual CFT.

Before we proceed, we will present a general result which will be very useful for us later.

3.2 General Formula for Ward Identities from CFT

One can derive a very useful and general formula for Ward identities in the presence of source terms. Consider two chiral quasi-primary fields and of conformal weights and , respectively, and the following general OPE

(3.12)

where is are chiral quasi-primary fields of weight and we have used the compact notation in case there are several fields with the same conformal weight. We are interested in expectation values of , but with insertions of source terms

(3.13)

where is a non-chiral source. Due to the insertion of , the vacuum expectation value will gain dependence. We can directly derive the following result

(3.14)

where we have used partial integration, the identity and finally

(3.15)

For illustrative reasons, let us take two simple examples. Let be the energy-momentum tensor and a primary field, we then have the following data from their OPE , , , and all other coefficients are zero. This leads to the identity

(3.16)

As a second example let us choose both fields to be the energy-momentum tensor . For this case we have the following OPE coefficients , , , , and . This leads to the following identity

(3.17)

As expected, this is just like the above result up to the central charge term. In the following we shall mainly use our result (3.14) the other way around, we will from the bulk derive the Ward identities then use (3.14) to find the OPE coefficients.

3.3 Holographic Operator Product Expansions and Superconformal Symmetries

The holomorphic conserved currents on the boundary can be organized into multiplets

(3.18)

where are bosonic fields of spin and and