Book Title

Book Title

[
July 27, 2019
Abstract

In these lectures we give an overview of the duality between gravitational theories of massless higher spin fields in AdS and large vector models. We first review the original higher spin/vector model duality conjectured by Klebanov and Polyakov, and then discuss its generalizations involving vector models coupled to Chern-Simons gauge fields. We proceed to review some aspects of the theory of massless higher spins, starting with the Fronsdal equations for free fields and moving on to the fully non-linear Vasiliev equations in four dimensions. We end by reviewing some recent tests of the higher spin/vector model duality at the level of correlation functions and one-loop partition functions.

Chapter 1 TASI Lectures on the Higher Spin - CFT duality

Simone Giombi]Simone Giombi

Department of Physics, Princeton University
Princeton, NJ 08544
sgiombi@princeton.edu

Contents

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1 Introduction

The aim of these notes is to provide a pedagogical introduction to some aspects of gravitational theories of massless higher spin (HS) fields, and in particular review recent progress in understanding their role in the context of the AdS/CFT correspondence.

The problem of constructing field theories describing the consistent propagation and interactions of HS fields () has a long history and is a highly non-trivial one. In the case of massless fields in particular, the possible interactions are greatly restricted by the higher spin gauge symmetry that must be present to decouple unphysical polarizations. In flat spacetime, powerful no-go theorems [1, 2], see e.g. [3] for a review, severely constrain the S-matrix in the presence of massless HS particles. Essentially, having massless HS fields results in higher conservation laws that are too restrictive to allow, under general assumptions, for a non-trivial S-matrix.

The flat spacetime no-go theorems can be circumvented if one assumes a non-zero cosmological constant, i.e. (A)dS backgrounds, where S-matrix arguments do not apply. Consistent cubic vertices of massless HS fields in (A)dS were explicitly constructed by Fradkin and Vasiliev [4] and, remarkably, a fully non-linear theory of interacting higher spins in (A)dS was found by Vasiliev [5, 6, 7, 8]. Let us summarize some important features of the theory:

  • Vasiliev constructed the exact non-linear equations of motion of the theory. They admit a vacuum solution which corresponds to the maximally symmetric space (A)dS.

  • In the simplest bosonic version of the model in AdS, the spectrum of fluctuations around the vacuum includes a scalar field with (this value corresponds to a conformally coupled scalar. It is above the Breitenlohner-Freedman bound [9]), and a tower of HS massless fields of all spins, . A minimal truncation to a spectrum involving only even spins is possible. It is essential for the consistency of the theory that the spectrum includes an infinite tower of HS fields.

  • While the original equations were written in 4d, generalizations to (A)dS for all have been constructed [10].

  • In all versions of the HS theory, the spectrum always includes a field, the graviton. Hence, HS gauge theories are in particular theories of gravity, which generalize the familiar Einstein theory by the inclusion of an infinite tower of massless higher spins.

  • The interactions among the higher spin fields involve higher derivatives; they carry inverse powers of the cosmological constant, and are singular in the flat space limit. The theory intrinsically lives in (A)dS.

  • Quantization of the theory is not fully understood at the moment, mainly because a conventional action for the theory has not been constructed yet.aaaInteresting work on the action principle in HS theories have appeared in recent years [11, 12, 13, 14, 15] (see [16] for earlier work), though it appears that an explicit action which at quadratic level reduces to the familiar free field actions is not available at present. Nevertheless, we believe that there should be an underlying, ordinary action in terms of the physical HS fields. For recent progress in constructing such actions perturbatively, see [17, 18]. However, one may speculate that due to the infinite dimensional HS symmetry, the theory may provide a UV finite model of quantum gravity. Some preliminary evidence towards this was recently given at one-loop level [19, 20].

While the Vasiliev HS theory was constructed several years before the discovery of the AdS/CFT correspondence [21, 22, 23], one can in fact see that, from the AdS/CFT viewpoint, it is natural that consistent theories of massless HS fields do exist: they have precisely the right structure to be dual to simple vector model CFTs at the boundary of AdS. Starting in Section 2 with a summary of the basics of HS symmetries in CFT, we move on to show how the various features of the HS theories listed above can be naturally understood from the CFT point of view. This will lead us to review the original higher spin/vector model conjecture of Klebanov and Polyakov [24] in Section 3 and 4, and its fermionic generalization [25, 26] in Section 5. In Section 7, we review the recent extensions [27, 28] of these conjectures to Chern-Simons gauge theories coupled to vector models, and the “3d bosonization” duality [29, 30] relating scalar and fermionic theories coupled to Chern-Simons. In Section 8, we review the Fronsdal equations for free massless HS fields, and then move on to describe the frame-like formulation of HS fields which is at the basis of Vasiliev non-linear equations. These equations, in the 4d case, are reviewed in Section 10. Finally, in Section 10.6 and 11, we review recent tests of the higher spin/vector model dualities at the level of 3-point correlation functions and vacuum partition functions.

While these notes are meant to be pedagogical, they are by no means comprehensive, and there are a number of topics that will not be covered. In particular, our focus will be mainly on the AdS/CFT dualities (and briefly on their higher dimensional generalizations). There is a large body of literature on the AdS/CFT version of the higher spin/CFT duality [31], see [32] for a review, which unfortunately we will not have time to cover.

2 Higher Spins from free CFT

To introduce the concept of HS currents and the corresponding symmetries, let us start with a very simple theory: a free massless scalar field

(0)

We keep for now the space-time dimension arbitrary, though the focus of these lectures will mainly be on the case . We assume Euclidean signature throughout. Of course, this model is a conformal field theory. As it is well known, it admits a conserved, traceless stress-energy tensor (see, for instance, [33])

(0)

This is an operator of spin 2 and conformal dimension , as one can see directly from the fact that a free scalar field has dimension . It is easy to explicitly verify (and left as an exercise) that this operator is conserved and traceless

(0)

up to terms that vanish by the equation of motion . Given a conformal Killing vector , which satisfies , one can construct from a conserved current , and from it a conserved charge in the standard way. The resulting conserved charges yield to the generators of the conformal group

(0)

which are in one-to-one correspondence with the conformal Killing vectors (there are + Killing vectors respectively for translations and rotations , and conformal Killing vectors for special conformal generators and dilatations ).

Now it turns out that this free CFT admits a much larger symmetry, which is an infinite dimensional extension of the conformal algebra: this is the HS algebra. The simplest and most explicit way to see this is to realize that the theory has a tower of HS operators, one for each even spin , which are conserved. These are bilinears in the scalar field carrying -derivatives, with the structure

(0)

where the curly brackets denote traceless symmetrization, so that the corresponding operator is totally symmetric and traceless, corresponding to an irreducible representation of of spin (note that odd spins are not present since we have a single real scalar field). The coefficients can be fixed by imposing conservation

(0)

For explicit calculations, it is often useful to employ an “index-free” notation by introducing an auxiliary polarization vector , which can be taken to be null, . In Euclidean signature, such null vector is complex, but this is no cause of concern for our purposes. Then, we may construct the index-free objects

(0)

Note that, since the polarization vector is null, trace terms are automatically projected out. One may “free” indices with the aid of the differential operator in the auxiliary polarization vector space [34, 35, 36]

(0)

Acting with this operator removes the polarization tensor while keeping track of the constraint , i.e. one has

(0)

and the conservation equation (2) may be compactly expressed as

(0)

The spin- operators (2) may be expressed in this language as

(0)

where we have encoded the coefficients into the function of two variables . Using the equation of motion , one can then show that the conservation equation (2) yields the differential equation

(0)

Equivalently, one may obtain the same equation by requiring that the operators (2) are primaries, i.e. they commute with the special conformal generators at . This is equivalent to conservation as a consequence of the conformal algebra and the fact that has spin and conformal dimension (since is a free field): this is the unitarity bound for a spin primary operator, and its saturation implies that is a conserved current [37].

The solution to (2) can be expressed in terms of Gegenbauer polynomials, see e.g. [35] for details. In terms of the representation (2), we may write (up to overall irrelevant normalization constant)

(0)

which defines the coefficients introduced above. Let us also mention that it is often convenient to package all HS operators into a single generating function (this is also natural from the point of view of the bulk HS theory, where the HS fields are packaged into a single “master field”, as we will see later). For instance, in a nice form of the generating function is given by [38]

(0)

The reader may verify that, expanding in powers of , this correctly reproduces the HS operators obtained above. One may also derive analogous generating functions in general dimension , but we will not give their explicit form here.

Given the conserved spin- currents , we can construct conserved charges which generate the corresponding symmetries. These can be obtained in a canonical way that generalizes the discussion reviewed above for the stress tensor. Given a spin-() conformal Killing tensor , bbbA conformal Killing tensor is a symmetric tensor satisfying . one can write down an ordinary conserved current

(0)

and from this the corresponding conserved charge in the usual way. For , this leads to the generators of the conformal group. For higher spins, we get further generators that are essentially higher derivative symmetries. They schematically act on the scalar field as

(0)

Note that there is one charge for each conformal Killing tensor . These can be constructed from tensor products of the conformal Killing vectors, and they organize in representations of the conformal group given by a Young tableaux with two rows with boxes (e.g. for spin 2, this is just the adjoint representation), see for instance [39] and [40]. The number of HS generators at each spin is then the dimension of this representation. In , for example, there are generators at spin . The algebra generated by commutators of these charges is the HS algebra. One of its distinguishing features is that it is intrinsically infinite dimensional; the only finite subalgebra is given by the spin 2 generators, corresponding to the conformal algebra. In general, commutators of charges produce charges of greater spins, and one needs the infinite tower to get a closed algebra. For example, the schematic structure of the commutator of spin 4 charges is

(0)

which shows that once a spin 4 charge is present, we must also have a spin 6 charge, and so on. This is of course clear from the explicit construction in terms of the free scalar CFT, where we see that we have an infinite tower of conserved currents, but one may proceed more abstractly and ask whether there can be other CFTs with (exact) higher spin symmetry. This question was answered by Maldacena and Zhiboedov [41], who showed that, assuming a CFT with an exactly conserved spin 4 current , then an infinite tower of conserved HS operators must be present in the theory, and all correlation functions of local operators coincide with those of a free CFT (the analysis of [41] was in ; the generalization to higher was studied in [42, 43, 44, 45]).

2.1 The free vector model

To move towards the AdS/CFT side of the story, we can consider a simple generalization of the above free scalar CFT. We take massless free scalars

(0)

with equation of motion

(0)

The model has now a global symmetry under which transforms in the fundamental, or vector, representation. We will therefore refer to this as the (free) vector model. There are now conserved higher spin currents carrying the indices

(0)

where the coefficients are the same as in the previous section. These operators may be decomposed into irreducible representations of

(0)

where are O(N) singlets; are in the symmetric traceless; and in the antisymmetric representation. The conformal stress tensor is the singlet . The spin 1 operator is just the current in the adjoint which corresponds to the global symmetry.

3 From CFT to AdS: higher spin/vector model duality

We now consider the following truncation of the vector model: we declare that we are interested only in correlation functions of invariant operators, i.e. we truncate the model to its singlet sector (as we will discuss more later, this can be done in practice by weakly gauging the symmetry). Then, there is a meaningful separation between “single trace” operators and “multi-trace” operators, where we borrow the terminology usually employed in matrix-type theories. In this vector model context, “single trace” just means that there is a single sum over the indices. Hence, the single trace operators are just the bilinears in . The full list of single trace primaries is thus exhausted by the singlet higher spin currents plus the scalar operator of dimension

(0)

This result is essentially equivalent to the Flato-Fronsdal theorem [46, 47]. The invariant operators containing more than two scalar fields are the equivalent of multi-trace operators. For instance, an operator of the schematic form

(0)

should be viewed as a double-trace operator. Its dimension is equal to the sum of the dimensions of its single trace constituents (and if interactions are turned on, it is equal to the sum of the single trace dimension plus corrections that are small in the large limit).

If we assume that the AdS/CFT correspondence holds in its most general form, we expect that this singlet sector CFT should be dual to some kind of gravitational theory in AdS, which becomes weakly coupled in the large limit. What should such AdS dual look like? According to the general rules of AdS/CFT

Thus, the CFT single trace spectrum (3) should match the single particle spectrum of the bulk dual (multi-trace operators such as (3) should be dual to multi-particle states in AdS). Conserved currents in a CFT are dual to corresponding gauge fields in AdS. The spin 1 and spin 2 examples of this are familiar in usual applications of AdS/CFT: a conserved spin 1 current is dual to a spin 1 gauge field in the bulk, and the conserved stress tensor is dual to the graviton. The same idea generalizes to all spins. Thus, for each conserved spin current we expect a corresponding massless HS gauge field in AdS

(0)

The global HS symmetry on the CFT side corresponds to a HS gauge symmetry in the bulk generated by a spin gauge parameter . In addition, the scalar operator should be dual to a bulk scalar field

(0)

where the mass of the bulk scalar is fixed by the familiar AdS/CFT relation in terms of the conformal dimension of the dual operator. In the free scalar CFT, we have and hence . As it turns out, (3) and (3) is precisely the spectrum of the so-called minimal bosonic Vasiliev higher spin theory in AdS [10]. In particular, the mass of the scalar field is not a free parameter and is dictated by the structure of the HS invariant equations.

The CFT and AdS single trace/single particle spectra clearly match, but what about interactions? In the CFT, the basic object of interest are correlation functions of primary operators. In the free CFT, these are easy to compute, using the explicit expressions for the currents given above and the free field Wick contractions. For instance, the 3-point function of the HS operators can be computed by the triangle diagram in Figure 1, where each line is a scalar propagator and the vertices include the appropriate derivatives. Explicit expressions for these correlation functions can be found for instance in [38, 48, 49] (see also [50] for higher point functions).

Fig. 1.: Three-point function of HS currents in the free scalar CFT. Solid lines represent free scalar propagators.

According to the AdS/CFT dictionary, these correlation functions should be matched to a dual Witten diagram calculation, as shown for instance in Figure 2 for the 3-point function.

Fig. 2.: Holographic 3-point function

Clearly, since the CFT correlation functions are non-zero, there must be non-trivial interactions of the HS gauge fields in the AdS dual theory. Therefore, from AdS/CFT point of view, it is not surprising that consistent interactions of massless HS fields in AdS should exist. And indeed, Vasiliev theory provides an explicit construction of such consistent interactions (even though, as mentioned earlier, Vasiliev equations were in fact written down before AdS/CFT).

The perturbative expansion in the bulk, controlled by the coupling constant denoted in Figure 2, is related as usual to the large expansion on the CFT side. We can see this as follows. If we normalize the currents so that they have 2-point functions of order one, i.e. , then the 3-point functions scale as

(0)

Therefore, we see that

(0)

Introducing Newton’s constant in the usual way as , we then see that the required scaling of the Newton’s constant with in the higher spin/vector model dualities is (we set the AdS scale to one here)

(0)

Note that this is different from versions of the AdS/CFT duality involving adjoint fields, where . In any case, it is clear that, as usual, the expansion on the CFT side is mapped to the perturbative expansion on the AdS gravity side, which runs in powers of the Newton’s constant.

A particularly interesting CFT correlation function is the 3-point function of the stress-tensor. This is given, to leading order in the bulk expansion, by a tree-level Witten diagram as in Figure 2, and it involves the cubic coupling of the graviton. One may ask whether this cubic coupling is perhaps the same as the one obtained from the ordinary two-derivative Einstein gravity

(0)

Let us specialize to the case of . It is known [33, 51, 49] that in a general 3d CFT the stress-tensor 3-point function is fixed by conformal invariance and conservation to be a linear combination of three structures

(0)

where the first two structure correspond to the stress-tensor 3-point functions in free scalar and free fermion theories (which yield independent tensor structures), and the third one is a parity odd tensor structure which does not arise in free theories, but can arise in interacting theories that break parity. Using AdS/CFT, the stress tensor 3-point function that follows from the ordinary Einstein gravity was computed in [52]. It turns out to be a linear combination of scalar and fermion structures.cccIn fact, it corresponds to the 3-point function of the stress tensor in the supersymmetric CFT. Therefore, the HS dual to the free vector model must have a cubic graviton coupling which is different from the ordinary Einstein gravity. The coefficients , in (3) can be changed by adding higher derivative terms to the Einstein action. So we learn that the HS dual should be a higher derivative theory of gravity, at the level of interactions. One can verify that this is indeed a feature of Vasiliev theory, as was briefly mentioned above: interactions involve higher derivatives.

From all the above arguments we see that a consistent theory of interacting HS gauge fields in AdS should exist, because it should provide the AdS dual of free CFTs. The known features of Vasiliev theory all agree with what we expect from CFT perspective. So it is very natural to conjecture [24] the higher spin/vector model duality

Note that, while the focus of these lectures is mainly on , this duality between free scalar vector model and HS gravity makes sense in any , and the corresponding non-linear HS theories are explicitly known [10].

3.1 Complex scalars and vector model

In the above discussion we focused on the theory of real scalars restricted to the singlet sector. Of course, we may also consider complex scalars, for which we can write the invariant Lagrangian

(0)

and restrict to the singlet sector. The single trace spectrum in this case then involves also odd spin currents

single trace ( singlets): (0)

For instance, for we have the familiar current . The dual HS theory should then involve HS gauge fields of all spins, one for each integer spin. In fact, this is precisely the spectrum of the bosonic Vasiliev theory in AdS [10], and the minimal theory involving even spins only (dual to the model) can be obtained from it by imposing a certain consistent truncation on the master fields. Let us mention that for the complex scalar theory (3.1) with even, one can also impose a singlet constraint [19]. In this case, the single trace spectrum includes one current for each even spin, and three currents of each odd spin. This spectrum is precisely dual to the Vasiliev theory based on the algebra [8], whose odd spin gauge fields transform in the adjoint of .

4 Interacting model and its AdS dual

The conjecture that the singlet sector of the free / scalar CFT is dual to Vasiliev HS gravity was first made precise in [24]. Earlier closely related work [53, 54, 55, 56, 57, 58, 59, 60, 61] focused on the correspondence between free CFTs with matrix-like fields and higher spin theories, stemming from the motivation of understanding the duality between SYM theory and type IIB string theory in the small gauge coupling limit. In free CFTs with matrix-like fields the single trace spectrum includes, similarly to the discussion of the previous sections, conserved HS currents which should be dual to massless HS gauge fields in AdS. However, in addition there is an exponentially growing number of single trace operators that are not conserved currents, and should be dual to massive fields in AdS. Vectorial CFTs provide much simpler realizations of the AdS/CFT duality, as elucidated in [24]: the spectrum of single trace operators in the singlet sector (3) just consists of the bilinears, and can be put in one-to-one correspondence to HS gravity theories of the Vasiliev type, as explained in the previous section.

While the duality between free vector model and HS gauge theory is already by itself a very interesting example of a potentially exact, non-supersymmetric version of AdS/CFT, a crucial observation made in [24] is that one can in fact easily generalize the duality to the case of large interacting vector models. Let us consider the standard model with quartic interaction

(0)

It is well-known that in , this model has non-trivial interacting IR fixed points. Indeed, the interaction is relevant for , and it triggers a RG flow from the free CFT in the UV to an interacting CFT in the IR (provided the bare mass term is suitably tuned to reach criticality). This RG flow is perturbative in the framework of Wilson-Fisher expansion. Working in , one finds the familiar one-loop beta function

(0)

and there is a weakly coupled IR fixed point at . One may then compute various physical quantities in the expansion, and provided sufficiently high order calculations are performed, one can obtain this way estimates in the physical dimension (), where the IR CFT is strongly coupled.dddThis typically requires some resummation procedure, see e.g. [62] for a review. The same interacting CFT may be also described, as it is well known (see e.g. [63] for a review), by the UV fixed points of the non-linear sigma model, which are weakly coupled in [64, 65].

A complementary approach that can be developed at arbitrary dimension , and is more natural for comparison with AdS, is the large expansion. A standard way to develop this expansion is based on introducing a Hubbard-Stratonovich auxiliary field as

(0)

where we have set the mass term to zero since we are interested in the CFT.eeeWe assume a regularization scheme such that mass is not generated if the bare mass is set to zero in the UV. Integrating out via its equation of motion , one gets back the original lagrangian. However, integrating out the fundamental fields generates an effective non-local action for

(0)

where we have assumed large and the subscript ‘’ denotes expectation values in the free theory. We have

(0)

In momentum space, the square of the propagator reads

(0)

and so from (4) one finds the effective quadratic action for

(0)

From this expression we see that for the induced kinetic term dominates in the IR limit compared to the bare quadratic term in (4), and hence the latter can be dropped at the IR fixed point. The conclusion is that, to develop the expansion of the critical IR theory, we may work with the action

(0)

where we have rescaled for convenience, so that its two-point function scales as , and acts as a coupling constant. From (4), we thus find the two-point function of in the IR to be

(0)

or, in coordinate space:

(0)

The large perturbation theory can then be developed using the propagator (4)-(4) for , the canonical propagator for and the interaction term in (4). From (4) we see that , which plays the role of in this description, behaves as a primary scalar operator of dimension (this value will of course be corrected at higher orders in the expansion). This means that under the RG flow, the dimension of the scalar operator has changed from to , as shown in Figure 3.

Fig. 3.: RG flow from free to critical model.

The quartic interaction in (4) may in fact be viewed as a particular example of the double trace deformations studied in [66], where one considers a CFT perturbed by the square of a single trace operator

(0)

The interacting model falls into this class, since it is a perturbation of the free CFT by the square of the operator. For the interaction in (4) is relevant and there is a flow to a new CFT in the IR. One can then show [66] that at large the dimension of goes from at the UV fixed point to at the IR fixed point.

The observation that the interacting model (4) may be viewed as a double-trace deformation of the free CFT allows to immediately deduce its AdS dual interpretation. Indeed, the AdS dual of the general double-trace flows of the type (4) is well-understood [67, 68]. In AdS/CFT, a scalar operator of dimension is dual to a bulk scalar field with boundary behavior (in Poincare coordinates )

(0)

and is related to the mass of the scalar field by the familiar relation (we set the AdS scale to one):

(0)

If we insist on unitarity, usually is the only possible solution (recall the unitarity bound ). However, for

(0)

both and are above the unitarity bound, and one may choose either one to specify the boundary condition (4) of the bulk scalar field. Then, according to the dictionary developed in [67, 68], one choice corresponds to the UV CFT, and the other to the IR CFT which sits at the endpoint of the RG flow triggered by the double-trace interaction (4).

The inequality (4) is precisely satisfied in the Vasiliev theory in AdS, where the scalar mass is given by . Therefore, both roots and are above unitarity and we can choose either one to quantize the bulk scalar field. The choice (in general, ) corresponds to the free CFT at the boundary, while to the interacting one. In other words, the critical model is dual to precisely the same Vasiliev theory, but with a different choice of boundary condition on the bulk scalar field.

But what about the higher spin currents? They are of course still present in the spectrum, but since the CFT is now interacting, the currents should not be exactly conserved anymore. Indeed, one finds that the divergence of the HS operators is now non-zero, except for , and at large takes the schematic form

(0)

where the coefficients can be determined using the equation of motion (see [69, 70] for the explicit result). Note that is a “double-trace” operator of the model, dual to a two-particle state in the bulk. This equation implies that the higher spin symmetry is only weakly broken at large . Indeed, using the fact that the 2-point function of a spin primary operator of dimension (recall that is the dimension of a conserved current, so denotes the anomalous dimension) is fixed by conformal invariance to be

(0)

the non-conservation equation (4) implies , and so the anomalous dimension to order may be extracted as [71, 72, 69, 70]

(0)

For instance, in , the explicit result takes the form [73, 70]

(0)

To summarize, in the critical theory, the dimension of the HS currents is only corrected starting at order

(0)

which means that the dual HS fields remain massless at the classical level, and receive masses through loop corrections [74].fffThe mass of a spin field is related to the dual dimension by , which implies . The leading correction should arise from a one-loop diagram in the bulk as the one depicted in Figure 4, where one of the loop lines involves the scalar field with boundary condition (the bulk-to-bulk propagator for the scalar field does depend on the choice of ). It was shown in [75] that, assuming no anomalous dimensions are generated by loops with the boundary condition, then the correct anomalous dimensions in agreement with the critical model would indeed be reproduced by the bulk loop diagrams, to all orders in . However, it remains to be shown that with the choice of boundary condition dual to free CFT the loop diagrams are indeed trivial (this should follow from the exact HS symmetry, but showing it explicitly by a bulk calculation is an open problem).

Fig. 4.: Witten diagram contributing to the one-loop correction to the HS current two-point function. The loop involves all fields in the spectrum, in particular the scalar field with boundary condition.

5 Fermionic CFT

So far we discussed the case of scalar CFT, but there is of course another free CFT that we can write in any dimension, namely a free massless fermion. More generally, let us take free massless Dirac fermions

(0)

This theory has a global symmetry under which transforms in the fundamental representation. Depending on the dimension , we may also impose a Majorana condition to obtain a CFT with symmetry. As in the case of the free scalar CFT, the model (5) admits an infinit tower of exactly conserved currents of the form

(0)

which are in the totally symmetric traceless representation of . Of course, one also has conserved currents in non-trivial representations of , but above we just wrote the singlet ones since we will be interested in projecting onto the singlet sector, as we have done in the scalar case. Note that the conformal dimension of the operators (5) is , as appropriate for a conserved current, since .

The explicit form of the currents is most conveniently given using an auxiliary null polarization vector, as described above in the scalar case. Imposing conservation (or the condition that is a primary), one finds that the conserved currents are given by

(0)

The reader may check as an exercise that for and this reproduces respectively the familiar current and the stress tensor . For , similarly to (2), one may also derive a simple generating function encoding all spins [27]

(0)

Let us now consider the truncation of the model to its singlet sector. If we restrict to the case, then it is not difficult to see that the HS currents (5), together with the parity oddgggIn , we may define “parity” by the sign reversal on all coordinates, . scalar operator

(0)

of dimension ( in general ), exhaust the spectrum of single trace operators

single trace ( singlets): (0)

In , it is also possible to impose a Majorana condition, so that the resulting theory has invariance, and the singlet sector involves only the even spin operators. Note that for general , there would be more operators in the single trace spectrum, because one can construct fermion bilinears involving products of more than one matrix (we will comment on this further in the next section).

The single trace spectrum (5) is very similar to the one in the free scalar theory, with the exception that the scalar operator is parity odd and has . The AdS dual should then be a HS theory which includes a pseudoscalar with , together with the tower of HS gauge fields of all (even) spins. It is also easy to see that the interactions in this theory must be different from the one in the dual to the free scalar, because the correlation functions of HS currents are given by different tensor structures in scalar and fermionic CFTs, as in the case of , eq. (3).

So we learn that there should be two inequivalent HS theories in AdS, with almost identical spectrum (except for the parity of the bulk scalar), but with different interactions. Indeed, as we will see later, there are precisely two parity invariant HS theories in AdS, which have the required spectrum and interactions. They are usually referred to as “type A” and “type B” theories, the former including a parity even scalar and the latter a parity odd one. The conjecture that the type B theory should be dual to the fermionic vector model was first made in [25, 26].

Note that in the type B theory, duality with the free fermion CFT requires that the bulk scalar is assigned the boundary condition. This corresponds to unbroken HS symmetry in this fermionic case. As in the scalar case, we expect that the alternate boundary condition corresponds to an interacting CFT related to the free one by a double trace deformation. This is just the familiar Gross-Neveu model

(0)

The interaction is irrelevant, but working in the large expansion one can show, by methods similar to the ones described above for the critical model, that there is a non-trivial UV fixed point where the scalar operator (which may traded by a Hubbard-Stratonovich auxiliary field) has dimension . We may refer to this CFT as the critical Gross-Neveu or critical fermion theory. The free CFT, with , sits now at the IR fixed point of the RG flow, as shown in Figure 5.

Fig. 5.: RG flow from the UV fixed point of the Gross-Neveu model to free fermions.

While the large expansion may be developed formally for any , it is clear that the UV fixed point is unitary only for , and in particular in the physically interesting dimension . Note that the UV fixed point may be also accessed perturbatively, at finite , in the expansion, where it is well known that the Gross-Neveu model has a weakly coupled UV fixed point. Let us also mention that the same critical fermion CFT admits a “UV complete” description as the IR fixed point of the “Gross-Neveu-Yukawa” model [76, 77, 63]

(0)

which has perturbative IR fixed points in that are expected to be equivalent to the UV fixed points of the Gross-Neveu model with quartic interaction (this can be checked explicitly at large by matching critical exponents computed in the two approaches). Note that the relation between the IR fixed points of the GNY model and the UV fixed points of the Gross-Neveu model is analogous to the relation between the Wilson-Fisher fixed points of the theory (4) and the UV fixed points of the non-linear sigma model in .

The AdS dual of the large critical Gross-Neveu model may be deduced analogously to the scalar case. It is the same type B theory dual to the free fermion CFT, but with the boundary condition assigned to the bulk pseudoscalar. In the interacting CFT, the HS currents are weakly broken at large and satisfy a non-conservation equation analogous to (4), which implies anomalous dimensions starting at order . In the bulk, one expects then the HS fields to acquire masses via loop corrections, as in Figure 4, when the scalar has boundary condition (but not when it has boundary condition).

6 Summary of parity invariant HS/CFT dualities

Let us summarize in Table 1 the AdS/CFT Higher Spin/Vector Model dualities [24, 26, 25] discussed so far.

Vasiliev HS type A Vasiliev HS type B scalar b.c. Free Scalar Critical Fermion (Gross-Neveu) scalar b.c. Critical Scalar (Wilson-Fisher) Free Fermion

Table 1.: Summary of AdS/CFT Higher Spin/Vector Model dualities [24, 26, 25]

In each of the cases shown in table, one may consider the or version of the vector models, which are respectively dual to the minimal (even spins only) and non-minimal (all integer spins) higher spin theories. An interesting feature of the non-minimal theories is that in addition to two possible boundary conditions one can impose on the bulk scalar, there is also a one-parameter family of conformally invariant boundary conditions one can impose on the bulk spin-1 gauge field [78, 79]. With the ordinary boundary condition () on a spin-1 gauge field in AdS, the bulk gauge field is dual to a conserved spin 1 current on the boundary, as we have assumed above. On the other hand, one may impose the alternate boundary condition (), which corresponds on the CFT side to gauging the global flavor symmetry. More generally one can impose a (parity breaking) mixed boundary condition, which corresponds to setting a linear combination of the “electric” field ( being the Poincaré radial coordinate) and the “magnetic” field to vanish at the boundary. With the mixed boundary condition, the dual CFT is obtained from the original one by gauging the global flavor symmetry and turning on Chern-Simons coupling at some level . The case corresponds to the ordinary boundary condition, while the “purely electric” boundary condition corresponds to . In the latter case, while one gauges the boundary flavor current, the kinetic term for the boundary gauge field is entirely generated from integrating out the matter fields at one-loop, corresponding to the case of three-dimensional critical QED [80, 81]. So, to summarize, the three-dimensional critical QED’s with bosonic or fermionic flavors, restricted to singlet sector, are holographically dual to type A or type B non-minimal Vasiliev theory, with the alternate boundary condition imposed on the bulk spin-1 gauge field, and possibly including a Chern-Simons term at level . By further imposing the alternate boundary conditions on the bulk scalar, one may also obtain the dual to the model, or the Gross-Neveu model coupled to a gauge field. Let us also briefly mention that it is possible to impose alternate boundary conditions, , on the higher spin fields as well: this corresponds to gauging the HS symmetry at the boundary, and the resulting theory is a conformal higher spin theory [82, 83, 84, 39].

In the table we have focused on the AdS/CFT case, which is the most well-understood so far, but it is natural to ask about higher dimensional generalizations of these dualities. In the free scalar case, as mentioned earlier, the spectrum of single trace operators is given by (3) in any , and this matches the spectrum of the known Vasiliev theory in AdS [10], which we may view as a generalization to all of the type A HS theory. In the free fermion case, on the other hand, for the single trace spectrum is more complicated than (5), as mixed symmetry representations of can appear [85, 86, 87, 88, 89, 90]. For instance, for the free fermion in restricted to the singlet sector, in addition to (5), there is an extra scalar and an extra tower of totally symmetric HS operators

(0)

and also a tower of operators in the mixed symmetry representation corresponding to a two-row Young diagram with boxes in the first row and 1 box in the second row

(0)

The dual HS theory in AdS, which should be viewed as a generalization of the type B HS theory, should then involve two scalars, two towers of totally symmetric HS fields, and one tower of mixed symmetry HS fields dual to (6). While one can write free equations for these fields in AdS, the full theory describing their interactions has not yet been constructed.

One may also ask if there is any interacting version of the duality in that one can obtain by changing boundary conditions of the bulk fields. In the scalar CFT case, where the bulk scalar field has , the alternate boundary condition is actually above unitarity for . This suggests the possibility of a unitary interacting vector model in , dual to Vasiliev “type A” theory in AdS with alternate boundary condition on the bulk scalar [91, 92, 20]. On the CFT side, since the quartic interaction is irrelevant for , this interacting CFT should be viewed as a UV fixed point of (4), whose existence can be seen formally in the large expansion [93, 94]. In [94], it was shown that the following model with scalars and invariant cubic interactions

(0)

posseses IR stable, perturbatively unitary fixed points in which provide a “UV completion” of the large UV fixed points of the model in . This proposal has passed various non-trivial checks [94, 95, 96]. These perturbative fixed points exist for , and are expected to be unitary to all orders in and expansions. However, non-perturbative effects presumably render the vacuum metastable via instanton effects. Understanding the counterpart of this instability from the point of view of the dual HS theory in AdS with boundary condition is an interesting open problem.

7 Chern-Simons vector models

One important feature of the HS/vector model dualities discussed so far is that they involve a projection to the singlet sector of the CFT. This is essential for the matching of bulk and boundary spectra to work. Without this projection, we would have many more primaries in the CFT (for instance, itself) that do not have a counterpart in the Vasiliev HS theory. As mentioned earlier, a natural way to impose the singlet constraint is to weakly gauge the symmetry, and then consider the zero gauge coupling limit. This decouples the gauge field, but we still retain the constraint that only gauge invariant operators are physical.

In , there is a nice way to gauge the symmetry without breaking conformal invariance: we can couple the vector model to a Chern-Simons gauge field [27, 28]. For instance, in the case of the fermionic vector model, we consider the gauge theory

(0)

where is in the fundamental representation of the gauge group (similarly, one can consider the version). The singlet sector of the free vector model can be recovered by sending , which decouples the gauge field. Since we are interested in the large limit, we can take the ‘t Hooft limit

(0)

The singlet sector of the large vector model is then obtained by sending the ‘t Hooft coupling to zero. Similarly, one may couple the scalar vector model to a Chern-Simons gauge field

(0)

This construction naturally suggests that it should be interesting to study the more general theories with . As we will see below, it turns out that even when the gauge coupling is turned on, these models possess approximate HS symmetry at large and should be dual to (parity breaking) HS theories in AdS