Correlators in higher spin AdS_3 holography from Wilson lines with loop corrections

We study the correlators of the 2d W_N minimal model in the semiclassical regime with large central charge from bulk viewpoint by utilizing open Wilson lines in sl(N) 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_N 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 N=2,3.


Introduction
In [1] we computed three point functions with two scalar operators and a higher spin current in the 2d W N minimal model with 1/N corrections. The main aim of this paper is to give a bulk interpretation of the conformal field theory results. 1 The 1/N corrections (or 1/c corrections with c 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. 1 After completing this draft, we become aware of an interesting paper [2] appearing in the arXiv. The paper deals with loop corrections in two point Witten diagrams for higher spin theories on AdS d . Related previous works may be found in [3][4][5][6][7][8][9][10].
In [11] the 't Hooft limit with large N but finite λ = N/(k + N ) 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 c but finite N . The bulk description for the semiclassical regime is supposed to be given by Chern-Simons gauge theory based on sl(N )⊕sl(N ) dressed by perturbative matters [13][14][15]. The large c regime should be realized with a negative level k = −1 − N + O(c −1 ), thus the conformal field theory is non-unitary in the regime. 2 In [1] we evaluated correlators at the 't Hooft limit with 1/N corrections, but the results can be generalized for the semiclassical limit with 1/c corrections. We try to interpret the 1/c corrections in terms of sl(N ) Chern-Simons gauge theory. The W N symmetry of the minimal model is generated by higher spin currents J (s) (z) with s = 2, 3, . . . , N . We examine the following two and three point functions as For N = 2, the Chern-Simons theory reduces pure gravity theory as in [21,22], and in that case 1/c corrections have been examined in Virasoro conformal blocks [23] and the conformal weight of the scalar operator [24]. The validity of the method with N = 2 is formally supported by the analysis of conformal Ward identity [23,25]. 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 → 0. 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 N 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 N . For 1/c corrections, we mainly focus on the simplest examples with N = 2 and N = 3. We find that the three point functions from the Wilson line method are regularization scheme dependent at the 1/c 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 N = 2, the authors in [24] tried to reproduce the 1/c corrections in the conformal weight of the scalar operator from the bulk theory. They succeeded in doing so up to the 1/c order since it is regularization independent, but they failed at the 1/c 2 order due to the regularization issue. Adopting our prescription for regularization, we succeed in reproducing the 1/c 2 order corrections of conformal weight both for N = 2 and N = 3.
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 N minimal model of (1.1) at the semiclassical limit with 1/c corrections. In section 3, we explain our prescription to compute boundary correlators in terms of open Wilson lines in sl(N ) Chern-Simons gauge theory. We reproduce the minimal model results at the leading order in 1/c and describe our prescription to regularize divergences arising from loop diagrams. In section 4, we apply our method to the simplest case with N = 2. In particular, we reproduce the result in [24] for the two point function at the 1/c order and improve their argument for the next order in 1/c with the help of our analysis for the three point function. In section 5, we proceed to the N = 3 case and show that our prescription also works for this example. In section 6, we conclude this paper and discuss open problems.

W N 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 c but finite N in 1/c expansion. For this purpose we should describe the model in terms of c, N instead of k, N in (1.1). The parameter k is related to c, N as in 1/c expansion. Originally k is a positive integer, but here we assume an analytic continuation of k to a real value. See [14] for details on the issue. Using this relation, we can expand physical quantities in 1/c, and terms at each order depend only on N . The two point function is fixed by the symmetry as where h is the conformal weight of the scalar operator O h . The overall normalization can be set as 1 by changing the definition of O h . 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 instead of (2.2). The spectrum of primary states can be obtained with finite k, N by applying standard methods like coset construction as in [27]. The states are labeled as (Λ + , ω; Λ − ), where Λ + , ω, Λ − are the highest weights of su(N ) k , su(N ) 1 , su(N ) k+1 , respectively. The selection rule determines ω in terms of Λ + , Λ − , so we may instead use the label (Λ + ; Λ − ). 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 whereρ is the Weyl vector of su(N ). According to [15] (see also [13] for the original proposal), the state (0; Λ − ) corresponds to a conical defect geometry, and the generic state (Λ + ; Λ − ) is mapped to the geometry dressed by perturbative matters. In particular, the states (0; 0) and (f; 0) 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 (f; 0) is and we mainly deal with the operator O h + corresponding to the state in this paper. Expanding the conformal weight h in 1/c as the two point function becomes For the operator O h + we have which is obtained from the expression (2.5) with finite k, N . The problem will be whether we can reproduce correct the coefficients in front of log(z) and log 2 (z) 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 O h + with Virasoro conformal blocks. As seen below, we have effectively decomposed the W N vacuum block, which is fixed by the W N 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 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 for which the expression with finite k, N is given by [31] G ++ (z) = |F 1 (z)| 2 + N 1 |F 2 (z)| 2 . (2.11) Here the W N conformal blocks are 12) and the relative coefficient is (2.13) From the leading terms in z expansion, we can read off the conformal weights of the intermediate state. For F 1 (z) and F 2 (z), the intermediate states are found to be the identity and the state (adj; 0), respectively. Here adj represents the adjoint representation of sl(N ), and the conformal weight of the state is h(adj; 0) = (k + 2N )/(k + N ). This is consistent with the decomposition as f ⊗f = 1 ⊕ adj withf as the anti-fundamental representation of sl(N ). As discussed in [1], we only need to consider the W N vacuum block F 1 (z) in order to obtain the three point functions in (1.3). Therefore, we conclude that these three point functions are fixed by W N symmetry even with finite k, N . We obtain the three point functions with 1/c corrections by slightly modifying the analysis in [1]. We decompose the four point function (2.10) as (2.14) where V 0 (z) is the Virasoro vacuum block and V s (z) is the Virasoro block of spin s current. The coefficient C (s) is related to the three point function in (1. 3) as Since V s (z) start to contribute at the order of 1/c, we expand as The relevant part of the four point function (2.10) can be expanded in z and 1/c as where we have defined Solving the constraint equations from (2.14), we find for the leading order in 1/c. The first few examples are (C (2.20) The square of the three point function could be negative for N ≥ 3, 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 1/c, we can obtain 1/c corrections to the three point functions as well. At this order, the constraint equations for s = 3, 4, 5 are found to be .
From these equations, we obtain In particular, C . It is not difficult to extend the analysis for C

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(N ) Chern-Simons gauge theory and open Wilson lines. In subsection 3.2 we explain the representation of sl(N ) generators in terms of x-derivatives. In subsection 3.3, we compute the two and three point functions in (1.3) at the leading order in 1/c. In subsection 3.4, we give a prescription to regularize divergences arising from loop diagrams, and prepare for explicit computations for N = 2, 3 in succeeding sections.

Chern-Simons gauge theory and open Wilson lines
In three dimensions, pure gravity with a negative cosmological constant can be described by sl(2) ⊕ sl(2) 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 sl(N ) ⊕ sl(N ) Chern-Simons theory, whose action is given by Herek is the level of Chern-Simons theory and A,Ã are one forms taking values in sl(N ). The generators of sl(N ) can be decomposed in terms of the adjoint action of embedded sl(2) as Here g (s) denotes the spin (s−1) representation of sl (2), and we have adopted the principal embedding of sl (2). The generators in sl(2) (adjoint representation) and g (s) are denoted as V 2 n (n = −1, 0, 1) and V s n (n = −s + 1, −s + 2, . . . , s − 1), respectively. For the application to higher spin AdS 3 gravity, we need to assign an asymptotic AdS condition to the gauge fields. We use the metric of Euclidean AdS 3 as ds 2 = dρ 2 +e 2ρ dzdz, where the boundary is at ρ → ∞. In a gauge choice, we can set We have a similar expression forÃ but suppress it here and in the following. The configuration corresponding to AdS 3 background is given by a(z) = V 2 1 . The asymptotic AdS condition restricts the form of a(z) as [33][34][35][36] There are residual gauge symmetries preserving the condition (3.4), and a part of them generates W N symmetry near the AdS boundary. We can define classical Poisson brackets for the reduced phase space. Moreover, we can see that J (s) (z) in (3.4) generate the W N symmetry in terms of the Poisson brackets. At the classical level, the relation between the Chern-Simons levelk and the central charge c of the dual conformal field theory is given by the Brown-Henneaux one as [37] c = 6k .
See [33][34][35][36] for more details. At the leading order in 1/c, the rules for computing conformal blocks from the Chern-Simons theory with open Wilson lines were given in [16], see also [38] for N = 2. For the two and three point functions in (1.3), we use (3.6) Here hw and lw denote the highest and lowest weight states in finite dimensional representations of sl(N ), respectively, and P represents the path ordering. Moreover, we remove the ρ-dependence in the gauge field as A(z) = a(z) using a gauge transformation. We include 1/c corrections by extending the analysis in [23,24] for N = 2. At the leading order in 1/c, we treat the coefficient J (s) (z) in (3.4) as a function of z. At higher orders in 1/c, we regard J (s) (z) as an operator, and the expectation values of open Wilson lines are evaluated by using the correlators of J (s) (z), which are uniquely fixed by the W N symmetry.

Generators of sl(N ) algebra
In this subsection we explain our prescription to compute the matrix elements of sl(N ) algebra for evaluating the expectation values of open Wilson lines as in (3.6). We start with the simplest case with N = 2 and then extend the argument for generic N . For N = 2, there are several previous works in [23][24][25], and we start by clarifying the representation with x-derivatives in [23].
For two point functions we evaluate where |j, m belongs to the spin j representation of sl(2) with m = −j, −j + 1, . . . , j. We set the norm of these states as With these states, the sl(2) generators in the Wilson line are described by (2j +1)×(2j +1) matrices.
As in [23,25], it would be convenient to map the expression as then the sl(2) generators can be written as In [23], they proposed that the wave functions are given by We would like to give a derivation such that it can be extended for generic N . It is easy to obtain x|j, j = x 2j as a solution to the equation J + |j, j = 0. The others follow as The dual states j, m |x should satisfy (3.14) In particular, we have j, −j|x = δ(x) as in (3.11). The normalization is set to be a convenient value. We then apply the analysis to the case with generic N . A way to represent the generators of sl(N ) is using N × N matrices, and sl(2) generators V 2 n (n = −1, 0, 1) can be embedded as described, e.g., in appendix A of [13]. Then the other generators may be obtained as where (s − n − 1) of V 2 −1 are inserted. The fundamental representation of sl(N ) can be described by an N dimensional vector, which behaves as a spin (N − 1)/2 representation under the action of the embedded sl (2). Therefore, the description with N × N matrices can be given by (3.7) with j = (N − 1)/2 and open Wilson lines based on sl(N ) algebra. In this specific case, we can map the matrix representation to the one with x-derivatives using (3.10) and (3.15). In the representation with x-derivatives, the generators of sl(N ) should be given by [39] where where the first few expressions are In particular, we have N 3 = 1 for N = 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 |hw corresponding to the highest weight in the fundamental representation of sl(N ). As explained above, they can be expressed for (z 1 , z 2 ) = (0, z) as with h 0 = (1 − N )/2. Here the sl(N ) generators are written in terms of x-derivatives as in (3.17). We would like to treat them perturbatively in 1/k (or 1/c). Following the analysis in [24], we compute Integrating over z, we find  [23] for N = 2.
According to the current prescription, the two point function of O h + in (1.3) should be computed as where W h 0 (z) is evaluated by the correlators of J (s) in the W N theory. The leading order expansion in 1/k leads to as expected.
We are also interested in the three point functions in (1.3), which should be obtained as (3.26) The first non-trivial contributions come from the terms of order 1/k . At this order, we need to compute The normalization of higher spin currents in (3.4) corresponds to (see, e.g., [40]) (3.28) we find 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 h 0 = (1 + λ)/2 (or N = −λ), and this is related to the triality relation discussed in [14].

Prescription for regularization
The 1/c 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 z j diverge when two (or more) currents J(z i ) 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 N symmetry in terms of central charge c. In particular, we use the two point functions which reduce to (3.28) if we use the relation c = 6k in (3.5). At finitek, the relation of (3.5) should be modified, and corrections to higher spin propagators are automatically included by expanding in 1/c instead of 1/k, see [24] for some arguments. Divergence would arise at the coincident point z 2 = z 1 , and we need to decide how to regularize it. We introduce a regulator as by shifting the conformal weight of the higher spin current as s → s − . This choice is reasonable since it does not break the scaling symmetry. Analogously, we introduce the regulator to other correlators of higher spin currents J (s) by shifting the conformal wights of the current.
Introducing the regulator , integrals over z j become finite but have terms diverging at → 0. 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 J (s) (z) in (3.20). We introduce parameters c s such that (3.20) becomes (3.33) In terms of 1/c expansion, (3.22) is changed as where f (sn,...,s 1 ) n (z n , . . . , z 1 ) are given by (3.23). At the leading order in 1/c, c = 6k as in (3.5) and c s = 1. From the next order in 1/c, we shift the values of c s to remove divergences. Namely, we expand c s in 1/c as and absorb divergences in c (i) s (i = 1, 2, . . .) order by order. We conjecture that all divergences can be removed by these two ways of renormalization.
As explained above, we determine to remove divergences by properly choosing the "bare" values of parameters c s . However, we have still freedom to choose the terms independent of . Here we fix them such that the three point functions O h +Ō h + J (s) in (1.3) are reproduced from the Wilson line method as in (3.26). Since the three point functions can be fixed by the W N symmetry as shown in the previous section, we would say that the regularization scheme is determined by making use of the boundary symmetry. This is expected to fix all the ambiguities left, and other physical quantities should be predictable. In the following two sections, we examine concrete examples with N = 2, 3 and show that the 1/c corrections in the conformal dimensions of scalar operators can be reproduced from the bulk viewpoint up to the two loop level applying the prescription described above.

Correlators for N = 2
In this and the next section, we explicitly evaluate the loop corrections of the correlators in terms of open Wilson lines. We start with the simpler case with N = 2 and then move to a more involved one with N = 3. For N = 2, we can work with generic h 0 = −j, because the sl(2) generators in terms of x-derivatives as in (3.10) are available for the generic case as argued in subsection 3.2.
Two and three point functions with generic h 0 are obtained from analysis of conformal field theory as follows. For h 0 = −j, the 1/c correction of conformal weight is given as (2.6) with see, e.g., [24]. The 1/c expansion of the two point function is then (2.7). In the next subsection, we examine the two point function at the next leading order in 1/c. We reproduce the order 1/c result as h 1 in (4.1), and remove a divergence by renormalizing the overall factor of the open Wilson line. The three point function is fixed by the conformal Ward identity as in the current convention of J (2) given by (3.31). The c 0 order term follows from (3.30). In subsection 4.2, we fix the parameter c 2 introduced in (3.33) such that the 1/c order term is reproduced. In particular, this removes another type of divergence. With the regularization scheme, we reproduce the order 1/c 2 term as h 2 in (4.1) from two point function at the two loop order in subsection 4.3.

Two point function at 1/c order
and so on. Here f (2,...,2) n (z n , . . . , z 1 ) are defined in (3.23). Since the one point function vanishes as J (2) (z) = 0, the non-trivial contribution starts from W (2) h 0 (z) . The contribution corresponds to the one loop correction in the two point function of O h as in figure 1.
The integrals in W (2) h 0 (z) over z 1 , z 2 diverge, and we introduce a regulator as in (3.32), i.e., for spin two current. With the regulator, we obtain a finite result after the integration over z 1 , z 2 as Using (4.3) and c 2 = 1 + O(c −1 ), the above expression leads to up to the terms of order 0 and 1/c. We compare the above expression in (4.7) with the 1/c expansion of two point function in (2.7). We can see that the log(z) term correctly explains h 1 = −6h 0 (h 0 − 1) in (4.1) as shown in [24]. The expression in (4.7) has a term proportional to 1/ , which diverges for → 0. We can remove the divergence by changing the overall factor of the open Wilson line asW With the normalization, we have for → 0. In other words, we choose the -independent part such that the corresponding two point function has unit normalization as in (2.3).

Three point function
We have proposed that three point functions can be computed with open Wilson lines as in (3.26) and reproduced the tree level results as in (3.30). In this subsection, we examine the next leading order in 1/c. There are two types of contribution at the order as in figure 2 and we would like to examine them in turn.
The first one is from which is represented as diagram (a) in figure 2. Here we need to introduce the regulator to the three point function of spin two current. Our prescription is to shift the conformal weight from 2 to 2 − , so we use The integral becomes simpler by taking y → −∞ as up to the term of order 0 . The second one is from At the leading order in 1/c, the four point function is given by a sum over the products of the two point function as (4.14) Denoting up to the terms of O( 0 ). The integrals H Combining the results so far, we find The expression diverges for → 0, and we remove the divergence by properly choosing c Here a is an arbitrary constant, which shall be fixed shortly. With this choice of the parameter c 2 , there arises a contribution of order 1/c from the following term as up to the terms of order 0 . Here W h 0 (z) is given in (4.4). With this prescription, we have for → 0. Therefore, setting a = 1/2, we reproduce the expected result as (4.2) with h 1 = −6h 0 (h 0 − 1) in (4.1). In summary, we choose the parameter c 2 in (3.33) as in order to absorb a divergence from the one loop diagram and also reproduce the result from the conformal Ward identity.

Two point function at 1/c 2 order
In the previous subsections we have regularized divergences arising up to the one loop order. Our claim is that other quantities are predictable after the renormalization. Here we would like to examine the two point function at the two loop order. Generically two loop diagrams have one loop sub-diagrams, and there would appear non-local divergences from the sub-diagrams. After all one loop divergences are removed by renormalization procedure, we should have no non-local divergences at the two loop order. There would be local divergences remaining, which can be renormalized as for the one loop computations. As discussed in [24], two point function without proper renormalization does not reproduce the correct dependence on log(z) and log 2 (z) at the two loop order because of non-local divergences as 1/ log(z). Since now it is not expected to have such divergences after the renormalization, it should be possible to reproduce the correct shift of conformal weight even at the 1/c 2 order. We shall show that this is indeed the case in this subsection. We first evaluate the expectation value of the open Wilson line at the 1/c 2 order without renormalization, then we consider its effects. A contribution comes from W which is expressed as diagram (a) in figure 3. The integral is computed as Here we neglect the terms of O( ) and write down only the terms depending on log(z) or log 2 (z). In the rest of this subsection, we include only such terms. Another type of contribution arises from W h 0 (z) in (4.4). Defining we find 14;23 (z) = 360(h 0 − 1) 2 h 2 0 log(z) 5 13;24 (z) = − 720h 0 ((h 0 − 2)h 2 0 + 1) log(z) 5 These integrals correspond to diagrams (b), (c), (d) in figure 3, respectively. Summing over all contributions we find Therefore, a non-locally divergent term as 1/ log(z) remains, and the expression cannot be compared with (2.7). Now we include the effects of renormalization, namely, the change of overall factor as in (4.8) and the shift of parameter c 2 as in (4.21). These effects lead to an extra contribution as h 0 (z) +G (2) h 0 (z) + · · · , (4.27) whereG (2) h 0 (z) = The extra contribution can be evaluated as Thus in total we arrive at

Correlators for N = 3
In the previous section, we have illustrated our prescription by examining a simple example of sl(N ) Chern-Simons theory with N = 2. In this section, we extend the analysis to more involved case with N = 3. It is a rather straightforward generalization even though computations become complicated due to the existence of spin three current J (3) . In this paper, we adopt the representation of sl(N ) generators with x-derivatives as in (3.17), which is valid for arbitrary representation with h 0 = −j for N = 2 but only for the fundamental representation with h 0 = (1 − N )/2 for N ≥ 3. 4 With N = 3, the 1/c expansion of conformal weight is given by (2.6) with (2.8) as In the next subsection, we reproduce the conformal weight at the 1/c order as in h 1 above from the bulk viewpoint and renormalize open Wilson line. In subsection 5.2, we examine three point functions and fix the two parameters c 2 and c 3 in (3.33) to be consistent with symmetry. In subsection 5.3, we show that our prescription correctly reproduces the conformal weight at the 1/c 2 order as h 2 in (5.1).

Two point function at 1/c order
As for N = 2, we start by examining the two point function at the 1/c order. Since spin three current J (3) is involved along with spin two current J (2) , there are two types of corrections as h 0 (z) (5.2) at this order. The two are represents in figure 1 and figure 4, respectively. Here W (2) h 0 (z) is defined in (4.4) and Since we have already computed W (2) h 0 (z) as in (4.6), we just need to evaluate W (2) h 0 (z) . The prescription in (3.32) leads us to adopt with the shift of conformal dimension of J (3) from 3 to 3 − . Using this expression, we find up to the term of order 0 . Inserting h 0 = −1, we obtain (600 log(z) + 200) =1 + 1 c 32 + 64 log(z) + 82 3 (5.6) up to the terms of orders 0 and 1/c. In particular, the 1/c order correction of conformal weight is read off as h 1 = −32, which is consistent with (5.1). In order to remove the divergence at → 0 up to the 1/c order, we renormalize the Wilson line operator as which leads to the corresponding two point function of canonical form as in (2.3).

Three point functions
We move to three point functions with one conserved current. For N = 3, there are two choices of currents, i.e., spin two current J (2) and spin three current J (3) . We start by computing W −1 (z)J (2) (y) up to the 1/c order by following the previous analysis for N = 2. With the convention of J where W where we have used Furthermore, we need to consider a contribution of the form as Here W (1) Including the effect, we obtain for → 0 as in (5.20).

Two point function at 1/c 2 order
As for N = 2, we examine the two point function up to the 1/c 2 order and see whether we can reproduce the 1/c correction of conformal weight as in (5.1) after adopting the regularization. As before, we first evaluate the 1/c correction without renormalization and then include its effects.
There are contributions involving only spin two currents, which were already evaluated in (4.26). We find where we have used (5.30) and (5.7). Thus the log(z) and log 2 (z) dependent terms in the total contribution are = 3200 log(z) + 2048 log 2 (z) .
The coefficients in front of log(z) and log 2 (z) are precisely those in (2.7) with (5.1). We would like to emphasize again that there is cancellation among non-local divergences.

Conclusion and discussions
We have examined the two and three point functions ( As concrete examples, we have only examined Chern-Simons gauge theories based on sl(N ) with N = 2, 3. For N ≥ 4 we see no major difference even though computations would be quite complicated. For instance, we can reproduce h 1 in (2.8) by evaluating integrals in (3.24) up to the 1/c order and comparing the 1/c expansion of the two point function in (2.7). We consider the following integral as for s = 2, 3, . . . , 10. We conjecture that the above equality also holds for s > 10. Then, the 1/c order correction of scalar conformal weight for generic N can be read off as which matches h 1 in (2.8). For our purpose it is enough to work with the non-unitary duality, but other problems may require a unitary one, i.e., the 't Hooft limit of [11], see footnote 2. For the unitary duality, we should extend the analysis to the case with a higher spin algebra hs[λ], which is a gauge algebra of 3d Prokushkin-Vasiliev theory [12].
In particular, we would like to understand the precise relation between open Wilson lines and particles traveling in the bulk. An important open problem is to confirm our proposal that correlators in the 2d W N minimal model can be computed with open Wilson lines in sl(N ) Chern-Simons gauge theory including 1/c corrections. In particular, we have to extend the checks to higher orders in 1/c. We have conjectured that all divergences are removed by renormalizing the overall factor of the open Wilson line and the parameters c s in (3.33), but it is desirable to prove this claim. A different regulator was introduced in [24] by shifting 1/(z 2 21 ) a → 1/(z 2 21 + 2 ) a , but it breaks conformal symmetry. We can see that divergences from loop computations with this regulator cannot be absorbed by these changes, thus conformal symmetry in the regularization procedure should play an important role.
We have proposed our regularization prescription so as to be analogous to that for usual quantum field theory even though the precise relation is yet to be clarified. We offer to fix the interaction parameters by comparing them to "experimental data" that are obtained from dual conformal field theory in the current situation. Once they are fixed, then other quantities like the self-energy of the scalar propagator are claimed to be predictable. A particularly nice thing happens for N = 2. In this case, the 1/c order of the interaction parameter c (1) 2 was determined by using the information on h 1 in (4.1) through (4.2). Fortunately, h 1 can be obtained from the expectation value of the open Wilson line as in (4.7), therefore we do not need to refer to explicit boundary data and everything is computable in terms of bulk theory. Here we have only considered to the next leading order in 1/c, but it is natural to expect that the same is true for higher orders in 1/c as well. For N = 3, we fixed the 1/c order of the other interaction parameter c (1) 3 such that the equality in (5.20) is satisfied. Here the number 224/5 was borrowed from the W N minimal model. However, we believe that there should be a way to determine c 3 without referring to explicit boundary data, and it is an important open problem to find this out. We do not claim that our prescription is unique, and in fact a different one was adopted in [23] for N = 2. It is easier to see the physical meaning in our regularization procedure, but their prescription seems to be convenient for actual computations of conformal blocks.
In any case, it should be useful to understand the relation between different prescriptions.
In this paper, we have examined the duality of [11] in the semiclassical limit discussed in [13][14][15] with 1/c corrections, but it is also possible to extend the analysis to other examples. In particular, an N = 2 supersymmetric version of duality was proposed in [42], and the bulk description of its semiclassical limit was argued to be given by sl(N + 1|N ) Chern-Simons gauge theory [43]. See [44][45][46][47][48] for conical defect or black hole solutions in higher spin supergravity. We think that supersymmetric extension is important for the following two reasons. Firstly, it is usually expected that supersymmetry suppresses quantum effects, and it would enable us to examine higher order corrections in 1/c systematically. Secondly, supersymmetry helps us to study relations between higher spin gauge theory and superstring theory, and concrete examples have been discussed in [4,49,50] with N = 3 supersymmetry and in [51,52] with N = 4 supersymmetry. We would like to report on this extension in the near future.