Contents

YITP-17-92

Correlators in higher spin AdS holography

from Wilson lines with loop corrections

Yasuaki Hikida1 and Takahiro Uetoko2

Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,

Kyoto University, Kyoto 606-8502, Japan

Department of Physical Sciences, College of Science and Engineering,

Ritsumeikan University, Shiga 525-8577, Japan

We study the correlators of the 2d W minimal model in the semiclassical regime with large central charge from bulk viewpoint by utilizing open Wilson lines in Chern-Simons gauge theory. We extend previous works for the tree level of bulk theory to incorporate loop corrections in this paper. We offer a way to regularize divergences associated with loop diagrams such that three point functions with two scalars and a higher spin current agree with the values fixed by the boundary W symmetry. With the prescription, we reproduce the conformal weight of the operator corresponding to a bulk scalar up to the two loop order for explicit examples with .

## 1 Introduction

In [1] we computed three point functions with two scalar operators and a higher spin current in the 2d W minimal model with corrections. The main aim of this paper is to give a bulk interpretation of the conformal field theory results.3 The corrections (or corrections with as the central charge) in the minimal model should be interpreted as loop corrections in the bulk gravity description. However, it is notoriously difficult to deal with divergences associated with gravitational loop diagrams in general. Applying holography, it is expected that boundary theory can define bulk quantum theory of gravity generically. For our case, the minimal model would determine the way to regularize these gravitational divergences, and we would like to show that this is indeed the case in this paper.

The 2d W minimal model has a coset description as

 su(N)k⊕su(N)1su(N)k+1 (1.1)

with the central charge

 c=(N−1)(1−N(N+1)(k+N)(k+N+1)). (1.2)

In [11] the ’t Hooft limit with large but finite of the minimal model is conjectured to be dual to the classical 3d Prokushkin-Vasiliev theory of [12]. Instead of the ’t Hooft limit, we consider the semiclassical regime with large but finite . The bulk description for the semiclassical regime is supposed to be given by Chern-Simons gauge theory based on dressed by perturbative matters [13, 14, 15]. The large regime should be realized with a negative level , thus the conformal field theory is non-unitary in the regime.4 In [1] we evaluated correlators at the ’t Hooft limit with corrections, but the results can be generalized for the semiclassical limit with corrections. We try to interpret the corrections in terms of Chern-Simons gauge theory.

The W symmetry of the minimal model is generated by higher spin currents with . We examine the following two and three point functions as

 ⟨Oh+(z1)¯Oh+(z2)⟩,⟨Oh+(z1)¯Oh+(z2)J(s)(z3)⟩ (1.3)

including corrections. Here is a scalar operator with conformal weight . The negative value of the conformal weight reflects the non-unitarity of the theory. At the leading order in , it was claimed in [16] that correlators or conformal blocks can be computed by the networks of open Wilson lines in Chern-Simons gauge theory.5 For instance, the expectation value of an open Wilson line computes the two point function . Roughly speaking, the open Wilson line corresponds to a particle running in the bulk, which is dual to the boundary two point function. Furthermore, the three point function can be evaluated with the extra insertion of the boundary current . The main aim of this paper is to interpret the corrections of the correlators (1.3) as loop corrections in the bulk computations with open Wilson lines. For , the Chern-Simons theory reduces pure gravity theory as in [21, 22], and in that case corrections have been examined in Virasoro conformal blocks [23] and the conformal weight of the scalar operator [24]. The validity of the method with is formally supported by the analysis of conformal Ward identity [25, 23]. See also [26] for a recent application.

During loop computations with open Wilson lines, we would meet divergences and a main issue in this paper is to propose a prescription to regularize the divergences. There are three main steps in the prescription. Firstly, we have to decide how to introduce a regulator to make integrals finite. We adopt a kind of dimensional regularization such that scaling invariance is not broken. Secondly, we have to remove the terms diverging for . Here we choose to shift parameters in the open Wilson line since we cannot remove divergences in the current setup with the shift of parameters in Lagrangian as for usual quantum field theory. Finally, we have to remove ambiguities arising from -independent parts in the shift of parameters. We offer a way to fix them so as to be consistent with the W symmetry of the minimal model.

It is easy to show that the Wilson line method reproduces the leading order results for correlators in (1.3) with generic . For corrections, we mainly focus on the simplest examples with and . We find that the three point functions from the Wilson line method are regularization scheme dependent at the order. Since the three point functions of the minimal model are fixed by the symmetry, we adopt a regularization such that the Wilson line results match the minimal model ones. For , the authors in [24] tried to reproduce the corrections in the conformal weight of the scalar operator from the bulk theory. They succeeded in doing so up to the order since it is regularization independent, but they failed at the order due to the regularization issue. Adopting our prescription for regularization, we succeed in reproducing the order corrections of conformal weight both for and .

The organization of this paper is as follows; In the next section, we summarize the results on two and three point functions (1.3) in the 2d W minimal model of (1.1) at the semiclassical limit with corrections. In section 3, we explain our prescription to compute boundary correlators in terms of open Wilson lines in sl Chern-Simons gauge theory. We reproduce the minimal model results at the leading order in and describe our prescription to regularize divergences arising from loop diagrams. In section 4, we apply our method to the simplest case with . In particular, we reproduce the result in [24] for the two point function at the order and improve their argument for the next order in with the help of our analysis for the three point function. In section 5, we proceed to the case and show that our prescription also works for this example. In section 6, we conclude this paper and discuss open problems.

## 2 WN minimal model in the semiclassical regime

In this section, we examine the two and three point functions (1.3) of the coset model (1.1) with large but finite in expansion. For this purpose we should describe the model in terms of instead of in (1.1). The parameter is related to as

 k=−1−N+N(N2−1)c+N(1−N2)(1−N3)c2+O(c−3) (2.1)

in expansion. Originally is a positive integer, but here we assume an analytic continuation of to a real value. See [14] for details on the issue. Using this relation, we can expand physical quantities in , and terms at each order depend only on .

The two point function is fixed by the symmetry as

 ⟨Oh(z)¯Oh(0)⟩=1|z|4h, (2.2)

where is the conformal weight of the scalar operator . The overall normalization can be set as by changing the definition of . This implies that the two point function is obtained only from knowledge of the spectrum. Throughout the paper, we only focus on the holomorphic sector, thus we may write

 ⟨Oh(z)¯Oh(0)⟩=1z2h (2.3)

The spectrum of primary states can be obtained with finite by applying standard methods like coset construction as in [27]. The states are labeled as , where are the highest weights of , respectively. The selection rule determines in terms of , so we may instead use the label . We should take care of the field identification in [28] as well. The conformal weight of the state can be obtained by coset construction [27] or Drinfeld-Sokolov reduction, see, e.g., [29, 30]. For instance, the latter gives the formula

 h(Λ+;Λ−)=|(k+N+1)(Λ++^ρ)−(k+N)(Λ−+^ρ)|2−^ρ22(k+N)(k+N+1), (2.4)

where is the Weyl vector of . According to [15] (see also [13] for the original proposal), the state corresponds to a conical defect geometry, and the generic state is mapped to the geometry dressed by perturbative matters. In particular, the states and correspond to the AdS vacuum, and a bulk scalar field on the background. Here we denote f as the fundamental representation. The conformal weight of the state is

 h+≡h(f;0)=(N−1)(k+2N+1)2N(k+N), (2.5)

and we mainly deal with the operator corresponding to the state in this paper.

Expanding the conformal weight in as

 h=h0+1ch1+1c2h2+O(c−3), (2.6)

the two point function becomes

 ⟨Oh(z)¯Oh(0)⟩=1z2h0[1−1c2h1log(z)+1c2(2h21log2(z)−2h2log(z))]+O(c−3). (2.7)

For the operator we have

 h0=1−N2,h1=−(N2−1)22,h2=−(N+1)2(2N(N+1)+1)(N−1)32, (2.8)

which is obtained from the expression (2.5) with finite . The problem will be whether we can reproduce correct the coefficients in front of and from the bulk viewpoint with open Wilson lines.

We also examine the three point functions in (1.3). In [1] we have evaluated the three point functions by decomposing the four point function of with Virasoro conformal blocks. As seen below, we have effectively decomposed the W vacuum block, which is fixed by the W symmetry in principle, and this implies that the three point functions can be fixed solely by the symmetry. Notice that the three point function with spin two current as

 ⟨Oh(z1)¯Oh(z2)J(2)(z3)⟩ (2.9)

is determined by the conformal Ward identity, and our conclusion may be regarded as a higher spin generalization.

We decompose the following four point function as

 G++(z) =⟨Oh+(∞)¯Oh+(1)Oh+(z)¯Oh+(0)⟩, (2.10)

for which the expression with finite is given by [31]

 G++(z)=|F1(z)|2+N1|F2(z)|2. (2.11)

Here the W conformal blocks are

 F1(z)=z−2h+(1−z)−2h++k+2Nk+N2F1(k+N+1k+N,−1k+N;−Nk+N;z), F2(z)=z−2h++k+2Nk+N(1−x)−2h+2F1(k+N+1k+N,−1k+N;2k+3Nk+N;z), (2.12)

and the relative coefficient is

 N1=−Γ(k+2N−1k+N)Γ(−Nk+N)2Γ(2k+3N+1k+N)Γ(−k−2N−1k+N)Γ(1−Nk+N)Γ(2k+3Nk+N)2. (2.13)

From the leading terms in expansion, we can read off the conformal weights of the intermediate state. For and , the intermediate states are found to be the identity and the state , respectively. Here adj represents the adjoint representation of sl, and the conformal weight of the state is . This is consistent with the decomposition as with as the anti-fundamental representation of sl. As discussed in [1], we only need to consider the W vacuum block in order to obtain the three point functions in (1.3). Therefore, we conclude that these three point functions are fixed by W symmetry even with finite .

We obtain the three point functions with corrections by slightly modifying the analysis in [1]. We decompose the four point function (2.10) as

 |z|4h+G++(z)=V0(z)+∞∑s=3(C(s))2Vs(z)+⋯, (2.14)

where is the Virasoro vacuum block and is the Virasoro block of spin current. The coefficient is related to the three point function in (1.3) as

 C(s)=⟨Oh+¯Oh+J(s)⟩⟨J(s)J(s)⟩1/2. (2.15)

Since start to contribute at the order of , we expand as

 C(s)=c−1/2[C(s)0+c−1C(s)1+O(c−2)]. (2.16)

The relevant part of the four point function (2.10) can be expanded in and as

 |z|4h+G++(z) (2.17) ∼1+1c∞∑n=1(1−N2)(−1n+NΓ(N)Γ(n)Γ(N+n))zn+1c2∞∑n=2f(n)czn+⋯,

where we have defined

 f(n)c(1−N2)2=1nn−1∑l=11l+Γ(n)Γ(N)N2Γ(N+n)(n−1∑l=0NN+l−1n−2−1N+11+N) −n−1∑l=1NΓ(N)Γ(l)(n−l)Γ(N+l)+(2N+11+N)1n. (2.18)

Solving the constraint equations from (2.14), we find

 (C(s)0)2=(1−N2)Γ(1+N)Γ(s−N)Γ(1−N)Γ(s+N)Γ(s)2Γ(2s−1) (2.19)

for the leading order in . The first few examples are

 (C(2)0)2=12(1−N)2,(C(3)0)2=16(1−N)2(2−N)(2+N). (2.20)

The square of the three point function could be negative for , and this is related to the fact that we are working in a non-unitary theory.

Examining the equation (2.14) at the next order in , we can obtain corrections to the three point functions as well. At this order, the constraint equations for are found to be

 f(3)c=f(2)c+2C(3)0C(3)1, f(4)c=f(2)c910+(1−N)28(1+N)2+1−N10(1+N)2+150(1+N)2+2C(3)0C(3)132+2C(4)0C(4)1, f(5)c=f(2)c45+(1−N)24(1+N)2+1−N5(1+N)2+125(1+N)2+2C(3)0C(3)1127+2C(4)0C(4)1⋅2 +2C(5)0C(5)1+(C(3)0)2[121−N1+N+67(1+N)+1849(1−N2)]. (2.21)

From these equations, we obtain

 C(3)1C(3)0=N3+3N2−3N−6N+2+1, C(4)1C(4)0=N3+29N24+3N2+1892(N−3)−8N−2+4740(N−1)−310(N−1)2 −2740(N+1)−310(N+1)2−6N+2−36N+3+1614, (2.22) C(5)1C(5)0=N3+155N212+29N2+800N−4−180N−3+257(N−1)−257(N+1)−6N+2 −36N+3−120N+4+3592.

In particular, for . It is not difficult to extend the analysis for at least up to by directly applying the analysis in [9].

## 3 Preliminaries for bulk computations

In this section, we explain our prescription to compute the two and three point functions (1.3) from bulk theory. In the next subsection, we introduce sl Chern-Simons gauge theory and open Wilson lines. In subsection 3.2 we explain the representation of sl generators in terms of -derivatives. In subsection 3.3, we compute the two and three point functions in (1.3) at the leading order in . In subsection 3.4, we give a prescription to regularize divergences arising from loop diagrams, and prepare for explicit computations for in succeeding sections.

### 3.1 Chern-Simons gauge theory and open Wilson lines

In three dimensions, pure gravity with a negative cosmological constant can be described by Chern-Simons gauge theory [21, 22]. As a natural extension, we can construct a higher spin gauge theory using Chern-Simons theory based on a higher rank gauge algebra [32]. We are interested in Chern-Simons theory, whose action is given by

 S=SCS[A]−SCS[~A],SCS[A]=^k4π∫tr(A∧dA+23A∧A∧A). (3.1)

Here is the level of Chern-Simons theory and are one forms taking values in . The generators of sl can be decomposed in terms of the adjoint action of embedded sl as

 sl(N)=sl(2)⊕(N⨁s=3g(s)). (3.2)

Here denotes the spin representation of sl, and we have adopted the principal embedding of sl. The generators in sl (adjoint representation) and are denoted as and , respectively.

For the application to higher spin AdS gravity, we need to assign an asymptotic AdS condition to the gauge fields. We use the metric of Euclidean AdS as , where the boundary is at . In a gauge choice, we can set

 A=e−ρV20a(z)eρV20dz+V20dρ. (3.3)

We have a similar expression for but suppress it here and in the following. The configuration corresponding to AdS background is given by . The asymptotic AdS condition restricts the form of as [33, 34, 35, 36]

 a(z)=V21−1^kN∑s≥21NsJ(s)(z)Vs−s+1,Ns=tr(Vs−s+1Vss−1). (3.4)

There are residual gauge symmetries preserving the condition (3.4), and a part of them generates W symmetry near the AdS boundary. We can define classical Poisson brackets for the reduced phase space. Moreover, we can see that in (3.4) generate the W symmetry in terms of the Poisson brackets. At the classical level, the relation between the Chern-Simons level and the central charge of the dual conformal field theory is given by the Brown-Henneaux one as [37]

 c=6^k. (3.5)

See [33, 34, 35, 36] for more details.

At the leading order in , the rules for computing conformal blocks from the Chern-Simons theory with open Wilson lines were given in [16], see also [38] for . For the two and three point functions in (1.3), we use

 ⟨lw|W(z2;z1)|hw⟩,W(z2;z1)=Pexp(∫z2z1dza(z)). (3.6)

Here hw and lw denote the highest and lowest weight states in finite dimensional representations of sl, respectively, and represents the path ordering. Moreover, we remove the -dependence in the gauge field as using a gauge transformation. We include corrections by extending the analysis in [23, 24] for . At the leading order in , we treat the coefficient in (3.4) as a function of . At higher orders in , we regard as an operator, and the expectation values of open Wilson lines are evaluated by using the correlators of , which are uniquely fixed by the W symmetry.

### 3.2 Generators of sl(N) algebra

In this subsection we explain our prescription to compute the matrix elements of sl algebra for evaluating the expectation values of open Wilson lines as in (3.6). We start with the simplest case with and then extend the argument for generic . For , there are several previous works in [25, 23, 24], and we start by clarifying the representation with -derivatives in [23].

For two point functions we evaluate

 ⟨j,−j|W−j(z2;z1)|j,j⟩, (3.7)

where belongs to the spin representation of sl(2) with . We set the norm of these states as

 ⟨j,m|j,m′⟩=δm,m′. (3.8)

With these states, the sl(2) generators in the Wilson line are described by matrices.

As in [25, 23], it would be convenient to map the expression as

 ⟨j,−j|W−j(z2;z1)|j,j⟩=∫dx⟨j,−j|x⟩W−j(z2;z1)⟨x|j,j⟩, (3.9)

then the sl generators can be written as

 J+(=V2−1)=x2∂x−2jx,J3(=−V20)=−x∂x+j,J−(=V2+1)=∂x. (3.10)

In [23], they proposed that the wave functions are given by

 ⟨x|j,j⟩=x2j,⟨j,−j|x⟩=δ(x). (3.11)

We would like to give a derivation such that it can be extended for generic . It is easy to obtain as a solution to the equation . The others follow as

 ⟨x|j,m⟩∝(J−)j−m⟨x|j,j⟩=Γ(2j+1)Γ(j+m+1)xj+m. (3.12)

The dual states should satisfy

 ∫dx⟨j,m′|x⟩⟨x|j,m⟩=δm,m′, (3.13)

 ⟨j,m′|x⟩∝∂j+m′xδ(x). (3.14)

In particular, we have as in (3.11). The normalization is set to be a convenient value.

We then apply the analysis to the case with generic . A way to represent the generators of is using matrices, and sl(2) generators can be embedded as described, e.g., in appendix A of [13]. Then the other generators may be obtained as

 Vsn=(−1)s−1−n(n+s−1)!(2s−2)![V2−1[V2−1,...,[V2−1,(V21)s−1]]], (3.15)

where of are inserted. The fundamental representation of sl can be described by an dimensional vector, which behaves as a spin representation under the action of the embedded sl. Therefore, the description with matrices can be given by (3.7) with and open Wilson lines based on sl algebra. In this specific case, we can map the matrix representation to the one with -derivatives using (3.10) and (3.15). In the representation with -derivatives, the generators of sl should be given by [39]

 Vsn=s−1∑i=0(n−s+1)s−1−iai(s,h0)x−n+i∂ix, (3.16)

where

 ai(s,h0)=(s−1i)(−2h0−s+2)s−1−i(s+i)s−1−i (3.17)

with . The wave functions are precisely those in (3.11). The generators (3.16) with (3.17) are those of higher spin algebra hs for , and sl can be realized by hs with as an ideal, which removes generators with .

With the realization of generators, in (3.4) are computed as

 Ns=3√πΓ(s)(1−N)s−1(N+1)s−122s−2(N2−1)Γ(s+12), (3.18)

where the first few expressions are

 N2=−1,N3=15(N2−4),N4=−370(N2−4)(N2−9). (3.19)

In particular, we have for .

### 3.3 Correlators at the leading order in 1/c

In order to compute the correlators in (1.3), we need to consider the expectation values of open Wilson lines with corresponding to the highest weight in the fundamental representation of sl. As explained above, they can be expressed for as

 Wh0(z) =∫dxδ(x)Pexp[∫z0dz′(V21−1^kN∑s=21NsJ(s)(z′)Vs−s+1)]1x2h0 =Pexp[∫z0dz′(V21−1^kN∑s=21NsJ(s)(z′)Vs−s+1)]1x2h0∣∣ ∣∣x=0 (3.20)

with . Here the generators are written in terms of -derivatives as in (3.17). We would like to treat them perturbatively in (or ). Following the analysis in [24], we compute

 ddz[e−z∂xWh0(z)]=(−1^kN∑s=21NsJ(s)(z)e−z∂xVs−s+1ez∂x)[e−z∂xWh0(z)]. (3.21)

Integrating over , we find

 Wh0(z) =∞∑n=0(−1^k)n∫z0dzn⋯∫z20dz1N∑sj=2[n∏j=11NsjJ(sj)(zj)]f(sn,…,s1)n(zn,…,z1), (3.22)

where

 f(sn,…,s1)n(zn,…,z1) (3.23) =n∏j=1⎡⎣sj−1∑i=0(−2sj+2)sj−1−iai(sj,h0)(x+z−zj)sj−1+i∂ix⎤⎦1(x+z)2h0∣∣ ∣∣x=0,

see (3.3) of [23] for .

According to the current prescription, the two point function of in (1.3) should be computed as

 ⟨Oh+(z)¯Oh+(0)⟩=⟨Wh0(z)⟩, (3.24)

where is evaluated by the correlators of in the W theory. The leading order expansion in leads to

 ⟨Oh+(z)¯Oh+(0)⟩∣∣O(c0)=⟨Wh0(z)⟩∣∣O(c0)=1z2h0 (3.25)

as expected.

We are also interested in the three point functions in (1.3), which should be obtained as

 ⟨Oh+(z)¯Oh+(0)J(s)(y)⟩=⟨Wh0(z)J(s)(y)⟩. (3.26)

The first non-trivial contributions come from the terms of order . At this order, we need to compute

 ⟨Wh0(z)J(s)(y)⟩∣∣O(c0) =−1^kNs∫z0dz1f(s)1(z1)⟨J(s)(z1)J(s)(y)⟩ (3.27) =−1^kNs∫z0dz1Γ(2h0+s−1)Γ(2h0)(z−z1)s−1zs−11zs−1+2h0⟨J(s)(z1)J(s)(y)⟩.

The normalization of higher spin currents in (3.4) corresponds to (see, e.g., [40])

 ⟨J(s)(z1)J(s)(z2)⟩∣∣O(c)=−(2s−1)^kNs1z2s12. (3.28)

Using

 ∫z0dz1(z−z1)s−1zs−11(z1−y)2s=z2s−1(y−z)sys(Γ(s))2Γ(2s), (3.29)

we find

 (3.30)

The result is consistent with (2.19) in the convention of (3.28). In fact, it is the same as eq. (1.3) of [40] up to a factor if we set (or ), and this is related to the triality relation discussed in [14].

### 3.4 Prescription for regularization

The corrections of the two and three point functions in (1.3) can be evaluated from higher order contributions in (3.22) using the Wilson line method. However, integrals over diverge when two (or more) currents collide. Therefore, we need to decide how to deal with these divergences, and we explain our prescription in this subsection.

Let us start with the correlators of higher spin currents, which are uniquely fixed by the W symmetry in terms of central charge . In particular, we use the two point functions

 ⟨J(s)(z2)J(s)(z1)⟩=−(2s−1)cNs61z2s21, (3.31)

which reduce to (3.28) if we use the relation in (3.5). At finite , the relation of (3.5) should be modified, and corrections to higher spin propagators are automatically included by expanding in instead of , see [24] for some arguments. Divergence would arise at the coincident point , and we need to decide how to regularize it. We introduce a regulator as

 ⟨J(s)(z2)J(s)(z1)⟩=−(2s−1)cNs61z2s−2ϵ21 (3.32)

by shifting the conformal weight of the higher spin current as . This choice is reasonable since it does not break the scaling symmetry. Analogously, we introduce the regulator to other correlators of higher spin currents by shifting the conformal wights of the current.

Introducing the regulator , integrals over become finite but have terms diverging at . In the usual quantum field theory with a renormalizable Lagrangian, we can remove divergences by renormalizing the overall normalization of quantum fields and the parameters of interactions. In the current case, we offer to remove divergences in a similar manner. We first use the fact that the normalization of a two point function can be chosen arbitrarily by the redefinition of the operator. We remove a kind of divergence by changing the overall factor of the open Wilson line such that the corresponding two point function becomes the normalized one as in (2.3). We then notice that the three point interactions between two scalars and a higher spin field are governed by the coefficients in front of in (3.20). We introduce parameters