# An AdS Dual for Minimal Model CFTs

###### Abstract:

We propose a duality between the 2d minimal models in the large ’t Hooft limit, and a family of higher spin theories on AdS. The 2d CFTs can be described as WZW coset models, and include, for , the usual Virasoro unitary series. The dual bulk theory contains, in addition to the massless higher spin fields, two complex scalars (of equal mass). The mass is directly related to the ’t Hooft coupling constant of the dual CFT. We give convincing evidence that the spectra of the two theories match precisely for all values of the ’t Hooft coupling. We also show that the RG flows in the 2d CFT agree exactly with the usual AdS/CFT prediction of the gravity theory. Our proposal is in many ways analogous to the Klebanov-Polyakov conjecture for an AdS dual for the singlet sector of large vector models.

^{†}

^{†}preprint: HRI/ST/1011

## 1 Introduction

Two dimensional conformal field theories are probably the best understood amongst all quantum field theories. The local conformal symmetry described by the Virasoro algebra is in many cases powerful enough to lead to a complete determination of the operator spectrum, as well as to explicit formulae for the correlation functions. These theories thus give concrete instances of nontrivial fixed points of the renormalisation group, many of which have a realisation in statistical mechanical systems.

In higher dimensional CFTs, without the luxury of the local Virasoro symmetry, we have had to resort to other techniques to learn about nontrivial fixed points. One of the fruitful approaches has been to consider theories in which one has very many interacting degrees of freedom, the so-called large limit. For example, for vector models in dimensions with number of fields, one can infer the existence of nontrivial fixed points in the large limit. In fact, in this limit, the fixed points are perturbatively accessible, and one can compute, in a systematic expansion, anomalous dimensions and correlation functions. Thus the vector model exhibits the analogue of the Wilson-Fisher fixed point without having to resort to methods such as the expansion.

While it has always been surmised that the large limit is some kind of mean field like description, it was not until the advent of the AdS/CFT duality that one could make this idea precise (at least for gauge theories). This duality gives a classical description of the leading large behaviour. The unexpected feature was that this was in terms of a higher dimensional theory which typically involves gravity in an asymptotically AdS spacetime. In the case of matrix valued fields with degrees of freedom, the relevant description is believed to be in terms of a classical string theory. If one then takes the further limit of ultra-strong coupling (), the classical (super-)string theory reduces to Einstein (super-)gravity. This idea has had tremendous success in the last decade or so, and its repercussions are now even being felt in domains once far removed from string theory.

The connection of the large limit to gravity, however, remains very mysterious, and our current understanding is very much tied to the origins of the duality in D-branes and string theory (with all its additional baggage of supersymmetry and so on). One would like to have examples which are shorn of any unnecessary ingredients, and which give an idea of how this connection comes about. Such ‘distilled’ versions of the gauge-gravity duality are also interesting from the point of view of applications to realistic systems which often do not involve supersymmetry, for example. Moreover, since an Einstein gravity dual forces us into the regime of very strong coupling one would need to move away from this limit to describe systems with a coupling of order one. Generically, this would require us to be in a stringy regime with a large number of operators of finite anomalous dimensions. The technical complications of quantising strings in (asymptotically) AdS spacetimes prevents us from studying this regime easily.

An interesting via media is afforded by the so-called higher spin theories in AdS spacetimes [1]. These are theories containing (generically) an infinite number of massless interacting fields with spin (see [2] for an introduction). It has been suggested by several people [3, 4, 5, 6] that these theories might be relevant for the description of (a sector of) the weak coupling limit of large gauge theories. However, a striking and concrete conjecture was made in 2002 by Klebanov and Polyakov [7] who suggested that a particular higher spin theory on AdS might be exactly dual to the singlet sector of the interacting (as well as the free) vector model in dimensions at large . This is interesting for several of the reasons discussed above. The model has a close relation to various statistical mechanical systems. The interacting fixed point is nontrivial and yet not strongly coupled. Finally, it is a concrete duality which goes beyond the Einstein gravity limit and yet does not involve an entire stringy spectrum of operators. Recent calculations have provided non-trivial, interesting evidence for this conjecture, see in particular [8, 9, 10].

The aim of this paper is to propose another duality of this nature. In fact, we shall return to the well understood class of 2d CFTs and look for signatures of a higher dimensional classical gravity like description in a suitable large limit. This will give rise to a controlled environment in which to study the puzzle of the emergence of a gravity dual.

The large limit of various field theories in two and higher dimensions has been much studied. For some reason, however, this limit does not appear to have been much explored in the context of 2d CFTs (see however [11, 12, 13]), perhaps because they are solvable by other means. In this paper we study a family of minimal model CFTs which are given by coset WZW models

(1.1) |

where the denominator is the diagonal subalgebra, and the subscripts refer to the level of the current algebra. This family of CFTs includes in the special case of the usual coset description of the unitary minimal models of the Virasoro algebra [14]. Though the generalisation of these theories to arbitrary has been less studied compared to the case, several important facts about them are known. In particular, the spectrum of primaries and the fusion rules follow directly from those of the WZW models, and the characters can in principle be deduced. More interestingly, these theories are known [15] to possess a higher spin symmetry [16, 17, 18] (for a review see [19]), and the different minimal models (for finite, fixed and different values of ) are related to one another by an integrable RG flow. More details about the minimal models are explained in Sec. 2.

Here, we will look at these theories in the large limit. Specifically, we will define a ’t Hooft limit (see also [13]) in which we take

(1.2) |

It is interesting that the limit appears to be well defined and non-trivial. In particular, these theories behave like vector models, since their central charge equals and hence scales as . The discrete set of CFTs coalesce into a line labelled by the ’t Hooft coupling , where behaves like a free theory (of complex fermions), while is some sort of ‘strong’ coupling region. Notice that the coupling always remains of order one — an indication of the absence of a dual Einstein gravity regime. Furthermore, the spectrum of primaries simplifies remarkably in the ’t Hooft limit, in that the dependence of the conformal dimensions on the coupling becomes essentially linear. One of the nice features of this model compared to the vector model is the existence of the additional continuous parameter , which makes it closer to the supersymmetric gauge theories in higher dimensions. The details of the ’t Hooft limit are explained in Sec. 3.

In Sec. 4 we describe the higher spin theories on AdS which are dual to these large CFTs.
As was mentioned before, the CFTs have a higher spin symmetry, and so it is natural
that the bulk theory also possesses such a symmetry. In fact, it was recently pointed out in
[20, 21] that higher spin theories in AdS possess,
at the classical level, an asymptotic symmetry group which is indeed a two dimensional
-algebra. This generalises the observation of Brown-Henneaux for asymptotic
Virasoro symmetries in Einstein gravity on AdS [22].
Here we will consider a theory of higher spins, which contains, in addition to a large
tower of massless higher spin gauge fields, two complex scalar fields (of equal mass).
It is known that scalar fields can appear as additional matter fields in these AdS theories (precisely in pairs of equal mass). However,
their mass cannot be arbitrary since it is related to a parameter of the algebra
which plays the role analogous to
[23, 24, 25].^{1}^{1}1We thank Misha Vasiliev for this
remark. We will see that this parameter (and hence the mass) is indeed
mapped to the ’t Hooft coupling (1.2) of the CFT via

(1.3) |

In Sec. 5 we provide support for this conjecture. The first piece of evidence consists in matching the spectrum of the CFT with that of the bulk theory. In fact, a 1-loop computation for the higher spin gauge fields in AdS [26] had already revealed a match with the vacuum character of the algebra. In the specific higher spin theory being considered here one has additional scalar fields in the bulk, as well as additional primary fields in the CFT. We find highly non-trivial evidence that the two match for all values of the coupling . This requires the spectrum of dimensions in the CFT to take a special form which it obligingly does, but only in the large ’t Hooft limit.

The next piece of evidence consists of relating the behaviour of the CFTs under the RG flow with that in AdS. The lowest non-trivial primary operator has conformal dimension in the ’t Hooft limit. The ‘double trace’ operator is thus relevant, and it is known to be the operator that induces the RG flow to the nearby minimal model. Furthermore, one knows that it flows in the IR to an irrelevant operator of the form , where has dimensions (in the ’t Hooft limit) . Note that

(1.4) |

This precisely corresponds to what we have learnt from AdS/CFT. In fact, one of the bulk complex scalar fields that one has to add to the higher spin theory is precisely dual to , while the other, , is dual to . Though both fields in AdS have the same mass, there is a choice in how they are quantised [27, 28]. In fact, we have to quantise them in opposite ways such that they correspond to the two different dimensions for the operators dual to them. As was argued on general grounds in [29] (see also [30]), the RG flow takes one from the quantisation corresponding to in the UV to that for in the IR. In other words, the operator corresponding to , namely , must flow in the IR to the operator corresponding to , namely . This is thus in perfect agreement with the CFT result we mentioned earlier.

In Sec. 6 we outline a heuristic way to ‘derive’ this duality. This uses the fact that the bulk description of the higher spin fields is a Chern-Simons theory [31, 32]. One might therefore imagine the boundary theory to be a WZW theory. In fact, there is a very specific set of boundary conditions associated with requiring the spacetime to be asymptotically AdS — this is for example clearly explained in [33], see also [20, 21]. From the point of view of the WZW theory, these boundary conditions lead to a specific gauging (Hamiltonian reduction) which goes by the name of (classical) Drinfeld-Sokolov (DS) reduction. The bulk description in terms of the higher spin theories is, of course, classical and we do not have a quantum definition of the theory. What we propose is that the quantum version of the above classical DS reduction would define the quantum theory. This quantum theory is believed to be equivalent to the above coset model, provided that the levels are suitably identified [17, 34, 35, 36, 37]. In order to apply this line of reasoning to our situation the main open problem is to understand how to describe the scalar fields in this formulation. This approach should hopefully lead to a detailed understanding of the emergence of gravity and higher spin diffeomorphisms in AdS.

Finally, Sec. 7 contains concluding remarks. We have sequestered various details of the CFT and higher spin theories into three appendices.

## 2 A Family of Minimal Model Conformal Field Theories

In this section we describe the 2d CFTs which will be the key players on the field theory side. Here we outline some of their important properties. In the next section we will take the ’t Hooft large limit of these models.

### 2.1 The Minimal Models

The CFTs we are interested are the so-called minimal models. They are most easily described in terms of a coset [15]

(2.5) |

where, in order to obtain we consider . The central charge of the coset equals

(2.6) |

where is the dual Coxeter number of . For we have , and the central charge becomes

(2.7) |

where we have introduced the parameter that will sometimes be useful. Note that for this is just the familiar unitary series of the Virasoro minimal models that can be described by the above GKO construction with [14].

For the smallest value , (i.e. ), we have a theory with central charge which has an alternative realisation in terms of a theory of parafermions [38]. The other extreme case corresponds to (taking while keeping finite), where , and the symmetry algebra is equivalent to the Casimir algebra of the affine algebra at level [18, 15]. The Casimir algebra consists of all singlets in the affine vacuum representation. Since the affine algebra is at level one, it can be realised in terms of free bosons.

### 2.2 The Minimal Model Representations

The actual coset theory does not just involve the vacuum representation of the coset algebra
(2.5). The other states of the theory fall into highest weight representations of the coset
algebra. These are labelled by , where is a highest weight representation
(hwr) of , is a hwr of , and is a hwr of
^{2}^{2}2It is important to note though that the states in the coset do
not transform under any non-trivial representations of
.. Only those combinations are allowed where appears in the decomposition
of under the action of . The relevant selection rule is simply

(2.8) |

where is the root lattice of . In addition, there are field identifications: the two triplets

(2.9) |

define the same highest weight representation of the coset algebra, provided that is an outer automorphism of the affine algebra corresponding to , with , and , respectively. For , the group of outer automorphisms is , generated by the cyclic rotation of the affine Dynkin labels, i.e. the map

(2.10) |

where the first entry is the affine Dynkin label. In this notation, the allowed highest weight representations of at level are labelled by

(2.11) |

Note that the field identification (2.9) does not have any fixed points since acts transitively on the highest weight representations of at level .

### 2.3 Conformal Weights

It is easy to see that for any choice of highest weight representations , there always exists a unique , such that . Thus we may label the highest weight representations of the coset algebra in terms of unconstrained pairs . These pairs are still subject to the field identifications

(2.12) |

The conformal weight of the corresponding highest weight representation equals then

(2.13) |

where is the eigenvalue of the quadratic Casimir operator of — hence the -dependence — in the representation , see appendix B.1 for our normalisation convention. Here the representation is uniquely determined by the condition that . Furthermore, is a non-negative integer, describing the ‘height’ (i.e. the conformal weight above the ground state) at which the primary appears in the representation . Unfortunately, an explicit formula for is not available, but it is not difficult to work out for simple examples. Alternatively, one may use the Drinfeld-Sokolov description of these models (that is briefly reviewed in appendix B). In that language the highest weight representations are labelled by , and the conformal weights equal

(2.14) |

where is the Weyl vector of . For (the Virasoro minimal models), (2.14) reduces to the familiar formula

(2.15) |

with , . Here we have identified and .

In the following, the primary where is the fundamental
representation^{3}^{3}3Note that the representation of the affine algebra
has entries as in (2.10). Below we will mostly
drop the affine Dynkin label, and use a description in terms of the usual
() Dynkin labels for representations of .
with the trivial representation will play an important role.
Then (2.13) gives — in this case with

(2.16) |

where we have used that and , is the anti-fundamental representation, , and we get (again with ) . On the other hand, for the coset representation with

(2.17) |

An example with arises for the case where and , the adjoint representation. Then but , and we obtain

(2.18) |

where we have used that . Finally, the representation with and also has and , and the conformal weight is

(2.19) |

### 2.4 Fusion Rules and Characters

The fusion rules of the coset theory follow directly from the mother and daughter theory. Indeed, in terms of the triplets the fusion rules are simply

(2.20) |

where the fusion rules on the right-hand side are those of , and , respectively. Note that the fusion rules are invariant under the field identification (2.9). Since the fusion rules of the level one factor are just a permutation matrix, we can also directly give the fusion rules for the representatives as

(2.21) |

Closed form expressions for the characters of the minimal highest weight representations are known in terms of branching functions, see for example eq. (7.51) of [19]. However, these expressions are often difficult to evaluate explicitly. In the following we shall mainly be interested in the large limit of these models, in which case the low lying terms of the characters simplify. In particular, the vacuum character becomes in this limit

(2.22) |

since for the character is that of the Casimir algebra, see eq. (7.18) of [19]. For finite the corrections to this formula are a consequence of the null-vectors of the and factors in (2.5). For the case of the vacuum representation with , these appear first at height .

For the other characters there is a similar formula in terms of branching functions of the
affine level one representation to the horizontal (finite-dimensional) Lie algebra. However, as
far as we are aware, no simple explicit formulae for these branching functions are known.^{4}^{4}4We
thank Terry Gannon for discussions about this point. We have
worked out the first few branching rules for some small representations in appendix C,
and from that we can conclude that

(2.23) | |||||

(2.24) | |||||

(2.25) | |||||

(2.26) | |||||

These formulae will play an important role below.

### 2.5 The RG Flows

For fixed (finite) the models with different values of (or ) are related to one another by an
RG flow. This is most familiar for the Virasoro minimal models, for which the perturbing field
in the UV is the field with
[39].^{5}^{5}5Here, and in the following, we mean by the field
whose left- and right-moving conformal dimension is .
In the above conventions this field corresponds to ,
which has indeed , see (2.18). The RG flow that is induced by this
relevant perturbation connects the -th unitary minimal model in the UV, to the st in the IR.
In the IR, the perturbing field of the UV theory has become irrelevant. Indeed, it can be identified
with the field of the ’st minimal model [39]. The latter field
has conformal dimension in the ’st minimal model, and hence
can be identified with the field in that theory, see (2.19).

Similarly, the -field can be identified with . In the IR it flows to the -field of the ’st minimal model [40]. The latter field can be identified with the field in that theory. Note that this is compatible with the above since we have the fusion rules (for )

(2.27) |

Thus the normal-ordered product of the field with itself is the field, and similarly for the and field.

The generalisation to is believed to follow a similar pattern, see for example [41]. The relevant field of the ’th minimal model induces an RG flow, whose IR fixed point is the ’st minimal model. In the IR the perturbing field becomes irrelevant, and is to be identified with the field of the ’st model, i.e.

(2.28) |

The analogue of the field for is slightly more subtle. For , charge conjugation of is non-trivial, and there are therefore two fields that play that role. Indeed, the analogue of the fusion rules (2.27) are now

(2.29) |

where denotes the anti-fundamental representation of . Note that the conformal dimension of the field obviously equals that of , and similarly, . The analogue of the RG flow for the field is then

(2.30) | |||||

(2.31) |

As we shall see, these RG flows have a very nice interpretation in the bulk theory, following the general analysis of [29], see also [7].

## 3 The Large ’t Hooft Limit

With all of these preparations in place, we can now explain the large limit we shall be considering. If we take for constant , then it follows from (2.7) that , remembering that [11, 12]. In this limit, however, many important fields will have vanishing conformal dimension. For example, this will be the case for and , see (2.16) and (2.18).

It is therefore more interesting to consider the ’t Hooft like limit (see also [13]), where we take both , but keep the (renormalised) ’t Hooft coupling

(3.32) |

fixed. In this limit we get a family of CFTs with an effectively continuous central charge

(3.33) |

Note that the central charge scales as . In this sense, these theories behave like vector like models (whose degrees of freedom scale as ), rather than like gauge theory models (where the number of degrees of freedom scales as ).

Note that the ‘free case’, , corresponds to first taking , before
taking . At finite , the limit leads to a theory with
. In the large limit, we then expect this theory to have a description in terms of a
singlet sector of free complex fermions.^{6}^{6}6For large , we may ignore the difference between
and . The latter theory has a description in terms of
complex free fermions. This would be closely analogous to the free vector model
considered in [7]. In our context, the ‘singlet sector’ condition arises automatically
as a consequence of the coset construction, and does not have to be added in by hand.

Next we turn to the conformal weights. It follows from (2.14) that they become in this limit

(3.34) |

Note that the second term is typically at least of the same order, since . For example, for the fields discussed above, we find in this ’t Hooft limit

(3.35) |

as well as

(3.36) |

Obviously, this also agrees with the formulae obtained from the coset description, eqs. (2.16) – (2.19).

We should stress that there are ambiguities in how to define the large limit, and that we have implicitly made a choice in the above. For example, for , there exist at least two different (natural) limits of the unitary minimal models that have been considered in the literature [43, 44]. They lead to quite different limit theories: the spectrum of [43] is continuous, and the resulting theory seems to be similar to Liouville theory (see also [45, 46]), while the spectrum of [44] is discrete. Both are believed to lead to consistent correlation functions, and thus both seem to define viable large limits.

While these limits have only been analysed for , it is not difficult to see how their respective analogues would differ in our case. In order to explain this, let us consider the representations of the form , whose conformal dimension equals

(3.37) |

Since the Casimir is of order (for representations with a finite number of boxes in the Young tableau, the coefficient is one half times the number of boxes , see (B.97)), the conformal weight then behaves in the large limit as

(3.38) |

where is an integer. In the ’t Hooft limit, both and become large, and hence representations with an arbitrarily large number of boxes are allowed. There are now essentially two possibilities we can consider: we can either define the fields of the limit theory to be those associated to a family of representations with fixed , and then take — in this case the conformal weight will approach . Or, we consider fields, where, as we take , we also take , keeping only their ratio fixed. The latter prescription leads to a continuous spectrum (and is the analogue of the proposal of [43]), while the former leads to a discrete spectrum as in [44]. As will become clear below, the dual of the bulk gravity theory we are about to discuss corresponds to the second option, i.e. to a limit theory with a discrete spectrum. Indeed, the fields associated to the gravity dual are those that appear in finite tensor powers of the fundamental (and anti-fundamental) representation, and therefore will not grow with .

We should also note that is not the only limit point; for example, for the representations of the form the conformal dimension behaves as

(3.39) |

etc. Finally, we note that excitations of order are typically seen in symmetric orbifold CFTs arising from fractionalised momentum; the above behaviour could therefore be indicative of some string theory interpretation of our higher spin theory.

## 4 The Higher Spin AdS Dual

Now we want to switch gears and describe the dual gravity theory for the above large family of 2d CFTs; this will turn out to be a higher spin theory.

Higher spin field theories in three dimensions are relatively more tractable than their higher dimensional counterparts. Firstly, the massless higher spin fields themselves do not contain any propagating degrees of freedom (see e.g. [26] for a recent discussion). Secondly, one can (classically) truncate consistently to a finite number of them [47]. For instance, one can have theories in which one has massless fields of spin only, for any . Thirdly, there exists a Chern-Simons formulation of the classical action for these theories [48, 49]. For theories with a maximal spin , the Chern-Simons gauge group is (in Lorentzian signature) or in Euclidean signature. Thus the interacting theory of the higher spin fields can be expressed relatively compactly compared to the higher dimensional cases.

To be a bit more specific, the higher spin gauge fields can be expressed in terms of generalised vielbein and connection variables (generalising the familiar case of gravity)

(4.40) |

where is the spin of the gauge field. In a theory with maximal spin , all these variables for the fields with can be packaged together into two (or one , depending on the signature) gauge fields. This reflects the fact that all these fields are part of one single multiplet under the higher spin symmetry. The action is then given by

(4.41) |

where

(4.42) |

The Chern-Simons level (to be distinguished from the that appeared in the previous section) is related to the AdS radius by the classical relation

(4.43) |

In [21] (see also [20]) it was argued, using the above Chern-Simons formulation, that the theory with maximal spin has an asymptotic symmetry algebra. Already at the classical level one sees a centrally extended algebra with a central charge whose value was determined to be the same as the Brown-Henneaux result for Einstein gravity on AdS

(4.44) |

In appendix A we summarise, for completeness, the salient features of the frame-like formulation and its relation to the more conventional Fronsdal description in terms of symmetric tensor fields of higher rank.

So far we have only discussed pure higher spin theories.
In three dimensions one can also have, in addition to the higher spin fields,
separate matter multiplets (for a survey of these matters see [25]).
While in higher dimensions the matter fields always lie in
the same multiplet as the higher spin fields, in three dimensions
the matter multiplet is distinct and contains only scalar and/or fermion fields.
Moreover, the fields in the matter multiplet can be massive since they are not in the
same representation as the gauge fields. The mass is related to a deformation
parameter^{7}^{7}7The deformation parameter is sometimes
denoted by in the literature on higher spin theories [25]. We suggestively call it
here since it has exactly the same relation to the mass as the conformal dimension of the
boundary operator. of the higher spin algebra as

(4.45) |

Typically the matter multiplet contains four scalars, two with mass (4.45), and two
with . These scalars can additionally transform under a global symmetry group.
However, it is consistent to truncate this multiplet^{8}^{8}8We thank Misha Vasiliev for discussions
about this point. to just the two scalars
of mass (4.45), and this is what will be relevant for the following.
The interacting theory of
these scalars with the higher spin fields was constructed in [23, 24].
Finally, we should mention
that for generic , it is no longer possible to truncate the massless fields to a maximal spin.
Thus once we have added such fields (as we are about to do), we have to take the
limit.

We can now describe the higher spin theory we are interested in. It contains, in addition to the higher spin gauge fields, a matter multiplet containing two complex scalar fields of the same mass (4.45). We will take to lie in the window

(4.46) |

As is by now familiar from various AdS/CFT applications this implies that there are two alternative conformally invariant quantisations (which we denote by ) of these scalar fields. These correspond to the two different roots of (4.45) determining the asymptotic fall-off behaviour. We shall take one of the scalars, which we call , in the -quantisation and thus corresponding to . The other scalar, , will be taken in the -quantisation corresponding to . We will denote this particular one parameter family of theories by .

Our proposal can now be stated as follows. The minimal model CFT with ’t Hooft coupling is dual, in the large ’t Hooft limit, to the theory with the identification

(4.47) |

Note that both scalars have the same mass which is given by

(4.48) |

Before we begin to discuss this proposal further, let us note that both the CFT and the higher spin theory have the same symmetry. It then makes sense to identify the central charges; this leads to

(4.49) |

The bulk theory is only well-defined in the large limit (since we can only add massive scalar fields in this limit). Note that large means that is small (in units where ); thus the large limit is indeed the semi-classical limit, where one can trust the bulk description. For finite , we may view the CFT (in its full expansion) as the quantum definition of the higher spin theory.

In the next section we will present some non-trivial checks of the proposal at leading order in . We shall also give a heuristic derivation of some parts of the duality in Sec. 6.

## 5 Checks of the Proposal

In this section we shall subject the above proposal to essentially two consistency checks. First we shall explain in quite some detail (see section 5.1) that the spectrum of the two theories agrees. More specifically, we shall study the quantum 1-loop partition function of the higher spin theory, and see how it reproduces the full CFT spectrum in the ’t Hooft limit. This is quite a detailed consistency check, and it probes much of the structure of the CFT. The second consistency check concerns the RG-flow for which we observe a beautiful matching with the bulk analysis (section 5.2).

### 5.1 The Spectrum

In this section we want to calculate the 1-loop partition function of the higher spin theory and compare it to the full CFT spectrum. There are basically two parts to this calculation. For the higher spin fields, the 1-loop determinant was computed recently in [26] using the heat kernel techniques of [50]. For the answer is

(5.50) |

The higher spin theory we are interested in also contains two complex scalar fields, one corresponding to and one with , see (4.47). The 1-loop contribution from each complex scalar field is [51] (see also [50])

(5.51) |

where . Thus defining

(5.52) |

the total 1-loop partition function is

(5.53) |

Our claim is that this partition function agrees with the full CFT partition function of the model in the ’t Hooft limit!

We have so far not managed to find an analytic proof of this statement, but we shall give below what we regard to be highly non-trivial evidence in favour of this claim. Before we begin with the detailed checks, we should first explain intuitively why this could be true.

The first factor coming from can be identified with a (generic) vacuum character of the -algebra [26]. In our case, the character of the vacuum representation of the coset CFT is not generic since we consider the limit of rational theories at finite . However, as was explained in Sec. 2.4, see in particular eq. (2.22), the null vectors only modify the answer at height , and thus this modification does not play any role in the ’t Hooft limit. We therefore conclude that the contribution from the higher spin gauge fields — the first factor of — reproduces precisely the vacuum character from the CFT perspective.

The full CFT has obviously many additional states; indeed, the coset representations are labelled by the pairs , and the full spectrum (at finite and ) will include all such sectors. However, given the structure of the fusion rules, all states of the CFT can be obtained by taking successive fusion products of the generating fields

(5.54) |

where and are the fundamental and anti-fundamental representation of . Note that in the large limit, the two sectors corresponding to and (and similarly for and ) effectively decouple; at finite , the field obviously appears in the -fold fusion of with itself, but in the ’t Hooft limit we have to include both separately.

Now the key observation is that the conformal dimension of the first two fields in (5.54) is , while that of the second two fields is , see eq. (3.35). This suggests the identification

(5.55) |

i.e. that the product on the left gives the contributions of the fusion products involving multiple copies of and . Similarly, the other term should be identified with

(5.56) |

Putting all factors together then accounts for the full spectrum of the CFT. In the following we want to check this proposal in more detail. We shall consider the different pieces in turn.

#### The Fusion Powers of

The simplest consistency check is to consider the square root of (5.55), and confirm that it reproduces the states that appear in the fusion powers of , say. (Obviously, the analysis is identical for the fusion powers of .) Expanding out the first few terms in (5.51) with leads to

(5.57) | |||||

In order to identify this with characters, we also have to multiply the expression with the 1-loop determinant coming from the higher spin fields, (5.50). Then the low-lying terms of look like the sum of three representations with conformal dimensions , and , whose characters are

(5.58) | |||||

(5.59) | |||||

(5.60) |

respectively. Since , these characters agree then precisely, in the ’t Hooft limit, with the characters for the representation , see (2.23), the representation , see (2.25), and the representation , see (2.26), respectively. Here we have used that the conformal dimension of these fields, in the ’t Hooft limit, are

(5.61) |

as well as

(5.62) |

Note that these two representations are precisely the representations that appear in the fusion of with itself,

(5.63) |

in accordance with the fact that the terms that are proportional to second powers of correspond to two-particle states in the bulk.

#### Higher Orders

We would expect that this pattern continues for higher powers of and . While we have not yet attempted to prove this in general, there is one simple consistency check we have performed. Since has a non-trivial -dependence, the above can only work out if the -dependence is additive under taking tensor products. It follows from (3.34) that the -dependent term is proportional to . For representations that have a finite number of boxes in the Young tableau, the argument in (B.97) then implies that in the large limit

(5.64) |

where is the number of boxes in the Young tableau of . For representations that appear in finite tensor powers of the fundamental, the number of boxes is conserved under taking tensor products (for sufficiently large), and since the fusion rules do not mix and (that contribute with opposite sign), the statement follows.

#### Fusion Products of and

It is clear that the above analysis works identically for the other factor in (5.55), the one associated to fusion products of . However, in order to check (5.55), we also have to verify that the fusion products involving both and work out. The leading ‘mixed’ term arises by taking terms proportional to from both factors in (5.55); it is easy to see that their total contribution is precisely

(5.65) |

Taking into account the -descendants, this then implies that the character of the corresponding CFT representation should be (in the ’t Hooft limit)

(5.66) | |||||

Because these states are single-particle in each factor, they should arise from the tensor product (2.29), and hence transform in the representation of the coset algebra — the other representation that appears in this fusion product is the identity representation that is already accounted for by . This works out precisely (to the order to which we have done the calculation), because (5.66) agrees exactly with the character of , see eq. (2.24).

#### Fusion Products of and

It is fairly straightforward to see that the analysis works essentially identically for the terms in (5.56). The main difference is that we now have to determine the leading behaviour of the characters of the representations with being in turn , , , etc. It is not difficult to show that the leading behaviour of the character of is in fact the same as that for . For concreteness, let us concentrate on the case when . For , i.e. and , we have . Then the leading behaviour of the character is described by the branching function where we count the multiplicities with which the -representation appears in the level representation based on , since we have to look at those representations of the level factor that lead to the trivial representation when tensored with the ground state representation . However, this branching function is precisely what gives the leading part of the character. The other cases work essentially identically. Thus we conclude that the contribution from the left-hand-side of (5.56) accounts for the tensor products of and .

#### Fusion Product of and

Finally we also have to look at the terms that involve both contributions from (5.55) and (5.56). By the same argument as that leading up to (5.65) it is clear that the leading term of the gravity calculation is

(5.67) |

Since , and taking into account the -descendants, this then implies that the character of the corresponding CFT representation should be (in the ’t Hooft limit)

(5.68) | |||||

Let us first consider the case , in which case we should expect (5.68) to agree with the character of . For , , the level representation is . In this level affine representation we then have to look for those representations of the finite-dimesional algebra that have the property that

(5.69) |

Among the representations that appear in the affine level representation of , the only ones that satisfy (5.69) are