Conformal blocks from Wilson lines with loop corrections
Yasuaki Hikida Takahiro Uetoko
hep-th








YITP-18-06











Conformal blocks from Wilson lines with loop corrections










Yasuaki Hikida^a[E-mail: yhikida@yukawa.kyoto-u.ac.jp] and Takahiro Uetoko^b[E-mail: rp0019fr@ed.ritsumei.ac.jp]



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^aCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
^bDepartment of Physical Sciences, College of Science and Engineering,
Ritsumeikan University, Shiga 525-8577, Japan





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We compute the conformal blocks of the Virasoro minimal model or its W_N extension with large central charge from Wilson line networks in a Chern-Simons theory including loop corrections. In our previous work, we offered a prescription to regularize divergences from loops attached to Wilson lines. In this paper, we generalize our method with the prescription by dealing with more general operators for N=3 and apply it to the identity W_3 block. We further compute general light-light blocks and heavy-light correlators for N=2 with the Wilson line method and compare the results with known ones obtained using a different prescription.
We briefly discuss general W_3 blocks.



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§ INTRODUCTION



Two dimensional conformal field theory (CFT) with large central charge c is supposed to be dual to the semi-classical regime of three dimensional gravity on anti-de Sitter (AdS) space. Since holography is expected to provide a formulation of quantum gravity in terms of dual CFT, we would obtain some insights on the quantum aspects of gravity from 1/c corrections in dual CFT.
As a simple example, we examine the W_N minimal model, which has a coset description as

    su(N)_k ⊕su(N)_1 /su(N)_k+1

with the central charge

    c = (N-1) ( 1 - N(N+1)/(k+N)(k+N+1))   .

The dual bulk description was proposed to be the Chern-Simons gauge theory based on  sl(N) ⊕sl(N) Lie algebra <cit.>. In particular, we can study the quantum corrections of AdS_3 gravity in terms of the Virasoro minimal model given by the coset (<ref>) with N=2.
We consider the large c regime of the minimal model, which corresponds to a perturbative regime of Chern-Simons theory. 
The expression of central charge (<ref>) implies that k = - 1 - N + 𝒪(c^-1); thus, 
the large c regime of the model should be non-unitary.
Nevertheless, the holography has been argued to work
 <cit.>, and it should be useful at least for problems related to its symmetry algebra.

The basic quantities of CFT are the spectrum of operators and correlation functions or conformal blocks.
In particular, the conformal block decomposition of the correlator is an essential tool for recent developments on conformal bootstrap programs (see, e.g., <cit.>).
In terms of Chern-Simons theory, conformal blocks at the large c limit were proposed to be computed by the networks of open Wilson lines, and the proposal was confirmed for explicit examples <cit.>.
In this paper, we develop techniques to evaluate conformal blocks beyond the large c limit by applying recent works on loop effects for Wilson lines in  <cit.>, see also <cit.>.
The proposal of the map between conformal blocks and Wilson line networks in <cit.> can be easily extended  beyond the large c limit as in <cit.>. We evaluate the 1/c corrections of conformal blocks from the loop corrections of the Wilson line networks.
The computations are rather subtle,
and a main issue is how to regularize divergences associated with loop diagrams.
In the language of the usual Lagrangian formulation, how to deal with this kind of divergences was established. Since we are dealing with open Wilson lines instead of Lagrangian, it is quite non-trivial to formulate a general procedure. In our previous paper <cit.>, we offered a way to regularize divergences by renormalizing parameters in Wilson lines, which correspond to couplings between bulk scalars and a higher spin current, along with rescaling the overall factor of open Wilson lines.
Moreover, we showed that ambiguities related to the renormalization procedure can be removed by requiring that the correlators of boundary theory are consistent with the symmetry.
In this paper, we adopt the renormalization procedure in <cit.> for regularizing  divergences.


The primary operators of the W_N minimal model (<ref>) are labeled as (Λ_+ ; Λ_-), where Λ_± represent the highest weights of su(N) Lie algebra.
The operator with (f;0) (where f denotes the fundamental representation of su(N)) is dual to a bulk scalar field,
and the operators with (Λ;0) correspond to their composite fields <cit.>. On the other hand, the operator with (0;Λ) has its conformal weight of order c, and it is proposed to be dual to a conical defect geometry <cit.>.
The symmetry of W_N algebra for the model (<ref>) is generated by higher spin currents J^(s) with s=2,3,…,N. In <cit.>, we evaluated the correlators as

    ⟨𝒪̅_(Λ;0) (z_1) 𝒪_(Λ;0) (z_2) ⟩  ,   ⟨𝒪̅_(Λ;0) (z_1) 𝒪_(Λ;0) (z_2) J^(s) (z_3)⟩

with general Λ = 2j (where j represents the spin of representation) for N=2  and with  Λ = f for N=3.
In particular, we reproduced the 1/c corrections of conformal weight for operators dual to bulk scalars up to 1/c^2 order by adopting our regularization prescription.
[For N=2, the conformal weight of operator was reproduced up to 1/c order in <cit.> by applying a Wilson line method, which is possible without specifying any regularization prescription.]


In this paper, we extend the previous analysis and apply our Wilson line method to four point conformal blocks with N=2 and N=3. For N=2, we consider the correlator as

    ⟨𝒪̅_(Λ ; 0) (∞) 𝒪_(Λ ; 0) (1)   𝒪̅_(Ξ ; 0) (z) 𝒪_(Ξ; 0) (0)  ⟩

with (Λ , Ξ) = ( 2q , 2 j).
The conformal blocks were already examined by utilizing open Wilson lines in <cit.>. Since we adopt a different regularization prescription, our formulation is not the same as theirs. 
We can easily relate the two methods for the identity Virasoro block, so we focus on  general Virasoro blocks.
[We refer to the conformal block with the identity operator or its descendants exchanged as an identity block and to other conformal blocks as general blocks.] 
In <cit.>, they obtained the all-order expression in z for the general Virasoro block up to 1/c order except for the j,q-independent part. For the part, they obtained the first few terms in z expansion. We compute the general block with our formulation and obtain the all-order expression in z up to 1/c order including the j,q-independent part.

For N=3, we examine the four point conformal block (<ref>)
with setting Ξ = f for simplicity.
We explicitly compute the identity W_3 block using the product of two open Wilson lines and reproduce the CFT result. For the purpose we first extend the previous analysis for the two and three point functions (<ref>) by dealing with general Λ  for N=3.
The general W_3 blocks at the leading order in 1/c were obtained in <cit.> from the CFT viewpoint, and they were reproduced in <cit.> using the networks of the open Wilson line. We point out that our formulation combines these results in a quite nice way. 
We also analyze the 1/c correction of a simple non-identity W_3 block with (Λ , Ξ ) = (f , f).

We further consider the four point function with heavy operators 𝒪_(0;Λ) as

    ⟨𝒪̅_(0 ; Λ) (∞) 𝒪_(f ; 0) (1)   𝒪̅_(f ; 0) (z) 𝒪_(0; Λ) (0)  ⟩

with N=2. The heavy operator corresponds to a conical defect geometry as mentioned above, and we compute the correlator using an open Wilson line in the conical space. For quantum corrections to the open Wilson line, we adopt our formulation developed so far. For quantum corrections to the conical geometry, we make use of the results in <cit.> obtained by applying the coadjoint orbits of the Virasoro group in <cit.>.
In this way, we reproduce the previous results from CFT computation in <cit.>. 
The Wilson line method was used in <cit.> at the leading order in 1/c, where they used open Wilson lines for the heavy operators instead of conical spaces.
For general N, the computation with our method reduces to those in <cit.> at the leading order in 1/c.



The organization of this paper is as follows.
In the next section, we introduce the W_N minimal model and write down the expressions of conformal blocks examined in terms of Wilson lines in this paper.
In section <ref>, we explain our bulk method for conformal blocks particularly focusing on our prescription for regularizing divergences.
In subsection <ref>, we review our previous work in <cit.> on the two and three point functions (<ref>) for N=2, and in subsection <ref>, we extend the analysis by considering general light operators with N=3.
In section <ref>, we apply the method to compute conformal blocks from the bulk viewpoint.
In subsection <ref>, we obtain the identity W_3 block from the product of two Wilson lines and compare it to the CFT result.
In subsection <ref>, we examine general Virasoro blocks with our formulation and see the relation to previous results in <cit.>. We further discuss general W_3 blocks. 
In section <ref>, we apply our formulation to correlators with two heavy operators along with two light operators for N=2.
We conclude this paper and discuss open problems in section <ref>.
In appendix <ref>, we summarize CFT computations of four point conformal blocks in the W_N minimal model.
In particular, we obtain four point function of (<ref>) with Λ corresponding a rectangular Young diagram and Ξ = f.
As an application of  <cit.>, we compute the 1/c corrections of higher spin charges for the scalar operators using the method of conformal block decomposition.
In appendix <ref>, we include some technical details of sl(3) Lie algebras.
In appendix <ref>, we compute the 1/c correction of a simple non-identity W_3 block.
In appendix <ref>, we examine how correlators obtained from Wilson lines change under local conformal transformations.




§ W_N MINIMAL MODEL



The minimal model of W_N algebra can be described by  the coset (<ref>) with the central charge (<ref>), see, e.g., <cit.> for some details.
The primary states are labeled as (Λ_+;Λ_-) with Λ_± as the highest weights of  su(N) Lie algebra.
We denote n^±_i (n^±_N =0) as the numbers of boxes in the i-th row of Young diagrams corresponding to Λ^± and define power sums  as

    C_s (Λ_+ ; Λ_-) ≡2/ s∑_i=1^N [ α_+  (n^+_i  - |Λ_+|/N + ρ_i ) + α_-  (n^-_i  - |Λ_-|/N + ρ_i ) ]^s

by following <cit.>.
Here, |Λ_±| denote  the numbers of boxes in Λ_±, and α_± are given by

    α_+ = √(N+k+1/2(N+k))  ,   α_- = - √(N+k/2(N+k+1))  .

The components of the Weyl vector ρ are
ρ_i = (N+1)/2 - i .
The conformal weight of primary state (Λ_+;Λ_-) is then written as

    h (Λ_+;Λ_-) = C_2 (Λ_+;Λ_-) - c -N +1/24  .

The two point function of the operator (Λ_ +;Λ_-) is fixed by the symmetry as

    ⟨𝒪̅_(Λ_+ ;Λ_- )  (z_1)  𝒪_(Λ_+ ; Λ_-)  (z_2)  ⟩
    = 1/z_12^2 h (Λ_+; Λ_-)  .

The overall normalization can be set arbitrary by changing the definition of operator. Here and in the following, we only consider the holomorphic sector, but it is straightforward to include the anti-holomorphic sector.


We are interested in a regime with the large central charge (<ref>) but finite N, which implies that we should take

    k = - 1 - N + N (N^2 -1 )/c + N (1 - N^2) (1 - N^3)/c^2 + 𝒪 (c^-3)   .

In the regime, the conformal weight behaves as 𝒪(c^0) for Λ_- = 0
while 𝒪(c) for Λ_- ≠ 0. 
For a while, we focus on the case only with light operators (Λ_+ ; Λ_-) = (Λ ;0), and later, we examine correlators with Λ_- ≠ 0 as well.
For  N=2, we label the representation of  su(2) by spin quantum number j =n_1/2, where n_i = n_i^+ is defined above.
The conformal weight of the operator with j is given by

    h_j ≡ h (2j;0)
    = - j - 6 j(j+1)/c  - 78 j (j+1)/c^2  + 𝒪(c^-3)

in 1/c expansion.
For general N, we expand the highest weight in the fundamental weights ω_i as
Λ = ∑_i=1^N-1λ_i ω_i, where λ_i are the Dynkin labels.
As a simple but non-trivial example, we also study in some details the N=3 case,
where  the Dynkin labels (λ_1,λ_2) are related as n_1 = λ_1 + λ_2 and n_2 = λ_2.
The conformal weight of the operator with (λ_1,λ_2;0) is

    h (λ_1 , λ_2 ; 0)
       =- λ_1 -λ_2  -8 (λ_1 ^2+λ_1  λ_2 +3 λ_1  +λ_2 ^2+3 λ_2 )/c+ 𝒪 (c^-2)   .

For general N, the computation of correlators is simplified when Λ = n ω_m, which corresponds to the case with n_1 = ⋯ = n_m = n, n_m+1 = ⋯ n_N = 0.
The conformal weight of the operator with  Λ = n ω_m is

    h(n ω_m, 0)
        = 1/2 m n (m-N) + m n (N^2-1) (m-N) (n+N)/2 c+𝒪 (c^-2)   .

Expanding the conformal weight (<ref>) in 1/c as

    h (Λ ; 0) = h_0 (Λ ; 0) + h_1 (Λ ; 0) /c  + 𝒪(c^-2)    ,

the two point function (<ref>) becomes

    ⟨𝒪̅_(Λ ;0 )  (z_1)  𝒪_(Λ ; 0)  (z_2)  ⟩
     = 1/z_12^2 h_0 (Λ ; 0)  - 2 h_1 (Λ ; 0)  log (z_12)/c z_12^2 h_0 (Λ ; 0)   + 𝒪(c^-2)   .

Later we shall see that the bulk computations reproduce the shifts of conformal weight from the terms proportional to log(z_12).


The symmetry of W_N algebra is generated by higher spin currents J^(s)(z) with s=2,3,…,N.
In this paper, we adopt the convention for the higher spin currents in <cit.>.
In particular, the two point functions are given by

    ⟨ J^(s) (z_1) J^(s) (z_2) ⟩ = B_s/z_12^2s

with

    B_s = (1 - 2 s ) N_s c /6  ,   
    N_s =  3 √(π)Γ (s) (1 -N)_s-1 (N+1)_s-1/2^2s-2 (N^2 - 1) Γ(s+1/2)  .

For s=2,3, we have

    B_2 = c/2  ,    B_3 = (4 - N^2) c/6  ,   
    N_2 = -1   ,    N_3 = N^2-4/5  ,

and, in particular, the normalization for the spin 2 current is the same as the usual one for the energy momentum tensor, i.e., J^(2)(z) = T (z).

The three point function with a higher spin current J^(s) is fixed by the Ward identity as

    ⟨𝒪̅_(Λ_+ ; Λ_-) (z_1) 𝒪_(Λ_+ ; Λ_-) (z_2)  J^(s) (z_3) ⟩
            = w^(s) (Λ_+ ;Λ_-) ( z_12/z_13 z_23)^s
    ⟨𝒪̅_(Λ_+ ; Λ_-) (z_1) 𝒪_(Λ_+ ; Λ_-) (z_2)  ⟩  ,

where w^(s) (Λ_+ ; Λ_-) denotes the spin s charge of the operator (Λ_+ ; Λ_-).
The spin 2 charge is nothing but the conformal weight as w^(2) (Λ_+ ; Λ_-)  = h (Λ_+ ; Λ_-) in (<ref>). The expression of spin 3 charge
w^(3) (Λ_+ ; Λ_-) ≡ w (Λ_+ ; Λ_-) may be found in <cit.> as

    w (Λ_+ ; Λ_-) =  √( 2 N (N + 2) (1 - N^2) /((c-1) (2 + N)+3 N^2)) C_3 (Λ_+ ; Λ_-)   ,

where the overall factor is due to our convention for the spin 3 current.
For N=3, the spin 3 charge of the operator (λ_1 , λ_2 ; 0) is given by

    w(λ_1 , λ_2 ; 0) = λ_1 - λ_2/3 + 8 (15 λ_1+15 λ_2+13) (λ_1 -λ_2)/15 c + 𝒪 (c^-2)   .

For general N, the spin 3 charge of the operator (n ω_m  ; 0) is

    w(n ω_m  ; 0) = 1/6 m n (2 m^2-3 m N+N^2) 
            +
    m n (N^2-1) (2 m^2-3 m N+N^2) (3 n (N+2)+N (N+2)-2)/6 (N+2) c +
    𝒪(c^-2)

in 1/c expansion.


A main aim of this paper is to examine conformal blocks using open Wilson lines in the bulk theory. 
For N=2, we consider the four point function of scalar operators
in (<ref>) with (Λ,Ξ) = (2 q ,2 j), that is

    ⟨𝒪_q (∞) 𝒪_q (1) 𝒪_j (z) 𝒪_j (0) 
       ⟩  , 
         𝒪_l ≡𝒪_(2 l ; 0)  .

Here we have used 𝒪̅_l = 𝒪_l since 
the finite representation of su(2) is self-conjugate.
The identity Virasoro block has been evaluated using Wilson lines up to order 1/c^2  as <cit.>

    z^2 h_j𝒱_0 (z) =    1 + 2 h_j h_q/c z^2 _2 F_1 (2,2;4;z) 
        + 1/c^2[ h_j^2 h_q^2 k_a (z) + (h_j^2 h_q + h_j h_q ^2) k_b (z) + h_j h_q k_c(z) ] + 𝒪(c^-3)   ,

where  the expressions of k_a,k_b,k_c may be used as in (A.11) of <cit.>. Here, z^2  _2 F_1 (2,2;4;z) at the 1/c order corresponds to the global block
[The global part of Virasoro algebra is given by sl(2) algebra, and only the sl(2) descendants are considered as intermediate states for the global block.]
with the exchange of the energy momentum tensor. 
We do not compute the identity block with the Wilson line method since there is no significant difference from the previous computation in <cit.>.
The Virasoro block with the intermediate operator 𝒪_p was similarly obtained up to 1/c order as

    z^2 h_j𝒱_p (z)=    z^h_p_2 F_1 (h_p , h_p ;2 h_p ; z) 
       + 1/c[ h_j h_q f_a (h_p, z) + (h_j + h_q ) f_b (h_p, z) + f_c (h_p, z ) ] + 𝒪(c^-2)    .

All order expressions in z  for f_a and f_b and the first few orders in z for f_c were obtained in <cit.>. In subsection <ref>, we examine the general block, applying our method, and, in particular, we obtain all order expressions in z even for f_c.

For correlators in the W_N minimal model with general N, we may compute correlators by applying the Coulomb gas method as explained in appendix <ref>, see, e.g., <cit.>.
A conformal block is given by a choice of integration contour over the positions of screening operators inserted. Since it is non-trivial to identify proper integration contours,
[We are particularly interested in the identity W_N block, which appears in the s-channel decomposition of the four point function (<ref>). However, we cannot find out any references giving the s-channel expressions. The expressions in the t-channel or the u-channel were obtained in <cit.> when Λ or Ξ correspond to a degenerate representation, see also <cit.>.]
we consider the simple correlator 

    ⟨_(n ω_m ; 0)(∞) _(n ω_m ; 0)(1)  _(f; 0)(z) _(f; 0)(0)⟩  .

In this case, the Coulomb integrals reduce to a hypergeometric function _2 F_1 (a,b;c;z), and we can apply the analysis for the Virasoro minimal model, see, e.g., <cit.>.
The final result obtained in appendix <ref> is

    |z|^- 4 h (f;0) |1-z|^2 nmk+N+1/N(k+N)[ | _2F_1 (n + n/k+N,-m/k+N;-N/k+N;z ) |^2 
    
          + |  z^k+2N/k+N (1 -z)^m - N - n (k + N +1)/k+N_2 F_1  (1+m/k+N,-n+1-n/k+N;2 + N/k+N;z ) |^2  ]

with 

    ≡ -Γ(1+m/k+N) Γ(-n+1-n/k+N) Γ(1+N-m/k+N) Γ(n+1+N+n/k+N) Γ(-N/k+N)^2/Γ(n+n/k+N) Γ(-m/k+N) Γ(-n-N+n/k+N) Γ(-N-m/k+N) Γ(2+N/k+N)^2 .

The first and second terms correspond to the W_N blocks of identity and adjoint operators, respectively. In subsection <ref>, we examine the identity W_3 block up to the 1/c^2 order. In subsection <ref>, we give some comments on general W_3 blocks, and in appendix <ref>, we analyze the adjoint W_3 block with n=m=1 up to the 1/c order. 

The identity W_N block is a useful quantity even in the CFT perspective. 
For instance, we proposed a way to obtain the 1/c correction of spin s charge for the primary operator in the W_N minimal model (<ref>) by utilizing the Virasoro block decomposition of the identity W_N block in <cit.>. Since the W_N minimal model is a solvable model,
we know how to obtain the higher spin charges with finite N,k as in <cit.>. However, in practice, it is difficult to obtain their explicit expressions for larger s, and our method provides a simple way to obtain them including 1/c corrections.
In <cit.>, we have considered higher spin charges for operators only in the fundamental representation. In appendix <ref>, we extend the analysis by examining higher spin charges for the operator with Λ = n ω_m including 1/c corrections.
[See <cit.> for a recent work on higher spin charges for general operators in a supersymmetric model.]
Among others, we can reproduce the 1/c correction of spin 3 charge in (<ref>).



We also apply our approach to heavy-light correlators.
At the leading order in 1/c, general CFT correlators of this type were reproduced from the analysis of the open Wilson line in a conical space in <cit.>.
For the analysis of 1/c corrections, we may work on a simple form of the correlator as

    ⟨_(0; n ω_m )(∞) _(f; 0)(1) _(f ; 0)(z)  _(0; n ω_m)(0) ⟩  .

In this case, there is only one type of exchanged operator, and the Coulomb integrals can be carried out as (see, e.g.,  <cit.>)

    |1 - z|^- 4 h(f;0)| z |^ - 2  n m /N|_2 F_1 (-n, - m/N+k;- N/N+k; 1 - z )  |^2   .

We reproduce the CFT result for N=2 by applying our method for open Wilson lines and the analysis for conical spaces in  <cit.>.
The bulk approach should  be applicable also for general N, but we leave detailed analysis to future work.






§ METHOD FOR BULK COMPUTATION



In this section, we explain our method to compute correlators or conformal blocks from the networks of open Wilson lines. 
In the next subsection, we first introduce sl(N) ⊕sl(N) Chern-Simons gauge theory and the networks of Wilson lines corresponding to conformal blocks. We then explain  our prescription developed in <cit.> to remove divergences associated with loop diagrams. 
In subsection <ref>, we review our previous work in <cit.> on the two point function of a scalar operator and the three point function with two scalars and a spin 2 current for N=2.
The analysis can be extended straightforwardly for N=3 if the scalar operator belongs to the (anti-)fundamental representation of su(3).
In subsection <ref>, we extend our method by analyzing correlators involving general scalar operators.




 §.§ Chern-Simons theory and open Wilson lines



The bulk description of the W_N minimal model with large c was proposed to be given by sl(N) ⊕sl(N) Chern-Simons gauge theory <cit.>,
the action of which is

    S = S_CS [A] - S_CS [Ã]   ,   
    S_CS [A] = k̂/4 π∫tr( A ∧ d A + 2/3 A ∧ A ∧ A )   .

Here, k̂ is the level of the Chern-Simons theory, and A takes values in sl(N) Lie algebra.
[We neglect Ã throughout this paper, but Ã can be analyzed similarly.]
As a higher spin gravity, it is important to decompose the generators of sl(N) Lie algebra in terms of  embedded sl(2). We consider the principal embedding as

    sl(N) = sl(2) ⊕( ⊕_s=3^N g^(s))    .

Here, g^(s) denotes the (2s - 1)-dimensional representation of sl(2), which is generated by
V_n^s (n=-s+1,-s+2 , … , s-1).
A solution to the equations of motion can be written in a form as

    A = e^- ρ V_0^2 a(z) e^ρ V_0^2 dz + V_0^2 d ρ

by fixing a gauge. Here, ρ denotes the radial coordinate, and the boundary is located at 
ρ = ∞.
Moreover, z , z̅ are coordinates parallel to the boundary.
With the coordinates, the AdS metric is given by ds^2 = d ρ^2 + e^2 ρ dz dz̅, and the corresponding configuration of the gauge field is with a(z) = V_1^2.
As for the application to AdS gravity, we assign the asymptotic AdS condition, which restricts the from of the gauge field as
<cit.>

    a (z) = V_1^2 - 1/k̂∑_s = 2^N 1/N_s J^(s) (z) V^s_-s +1

with N_s in (<ref>).
The asymptotic symmetry near the AdS boundary is identified as the W_N algebra generated by J^(s).
The central charge of the algebra is

    c = 6 k̂

at the classical level, which is same as for the pure gravity in <cit.>.
With the coefficients of J^(s) in (<ref>), the normalization of currents is the same as that in  (<ref>) at the leading order in 1/c.


We would like to compute the correlators or conformal blocks of the W_N minimal model in terms of Wilson line operator

    W (z_f ; z_i) = P exp ( ∫_z_i^z_f d z    a(z)  )

in the sl(N) Chern-Simons theory.
Here we remove the ρ-dependence by a gauge transformation as A = a(z).
At the leading order in 1/c, it has been argued that n-point conformal blocks can be evaluated as <cit.>

    G_n(Λ_j|z_j) ≡⟨ S |  ∏_j = 1^n  W_Λ_j (z_0 ; z_j)   |hw⟩ _j  ,

where | hw⟩ _j denotes the highest weight state in the representation Λ_j of sl(N).
Moreover, ⟨ S | belongs to a singlet representation in ⊗_j=1^n Λ_j.
There are several ways to construct a singlet state, and a choice of singlet leads to one of conformal blocks, see <cit.> for more explanations and explicit examples.
The n open Wilson lines are connected at a point z_0, but the expression should be independent of z_0. For n=2, we may set z_0 =z_2,
[If we instead set z_0 =z_1, then we obtain G_2 (Λ,Λ̅| z_1 , z_2) =  ⟨lw | W_Λ̅ (z_1 ; z_2) | hw⟩ with | hw⟩ and ⟨lw | in the representations Λ̅  and Λ, respectively.
]
then, the expression of (<ref>) becomes

    G_2 ( Λ, Λ̅| z_1 , z_2) =  ⟨lw | W_Λ (z_2 ; z_1) | hw⟩  .

Here ⟨lw | represents the lowest weight state in the representation Λ̅, which is the conjugate of Λ.


For evaluating quantum effects in the Wilson line network (<ref>), we should integrate over gauge fields in the path integral with the insertions of Wilson line operators. Following <cit.>, we evaluate the expectation values by using the correlators of J^(s)(z) in (<ref>) with (<ref>) in terms of the theory with W_N symmetry as in (<ref>). Here the normalization (<ref>) is set in terms of the central charge c, so it is regarded as the full propagator after including loop corrections, see <cit.> for the arguments. We also need to integrate over z along the Wilson line in (<ref>). However, the integration would diverge when at least two of J^(s)(z) collide at the same point. As in <cit.>, we introduce a regular ϵ and replace (<ref>) by

    ⟨ J^(s) (z_1) J^(s) (z_2) ⟩ = B_s/z_12^2 s  - 2 ϵ  .

A merit of this choice is that the regulator does not break the scale invariance. The integration over z_i would diverge when ϵ→ 0, and we remove the divergence by renormalizing the overall normalization of Wilson lines and  parameters corresponding to interactions as in <cit.>. We denote the parameters as 

    c_s = 1 + 1/c c_s^(1) + 𝒪 (c^-2)   ,
 
and insert them into the Wilson line operator as

    W (z_f;z_i)
    = P exp[ ∫_z_i^z_f dz (  V_1^2 - 6/c∑_s=2^N c_s/N_s J^(s) (z) V_-s+1^s) ]   .

We can remove some divergences at the 1/c order by using c_s^(1) and similarly for higher orders. There are still ambiguities for the terms of order ϵ^0, and we offer to fix them such that the boundary correlators are consistent with the W_N symmetry.




 §.§ Two and three point functions for N=2




In order to illustrate our method and fix notations, we review our analysis in <cit.> for the two and three point functions in (<ref>).
For N=2, it is convenient to work with the x-basis, which can be applied to the  general N case but with operators in the (anti-) fundamental representation. As in <cit.>, we consider the x-basis expression of (<ref>) as

    ⟨lw| W_h_0 (z_f ; z_i)  | hw⟩
     =
    ∫ d x⟨lw | x ⟩  W_h_0 (z_f ; z_i) ⟨ x | hw⟩  .

For brevity, we call the left hand side W_h_0 (z_f ; z_i).
With the basis, we can use

    ⟨lw | x  ⟩ = δ (x)   ,   ⟨ x | hw⟩ = 1/x^2 h_0  ,

where h_0 = -j for N=2 and h_0 = (1-N)/2 for general N. The generators are <cit.>

    V_n^s = ∑_i=0^s-1[ s -1;    i ](n - s + 1)_s - 1 - i (- 2 h_0 - s + 2)_s - 1 - i/(s+i)_s-1-i x^- n + i∂^i_x   ,

and, in particular, the sl(2) subalgebra is generated by

    V_1^2 = ∂_x   ,   
    V_0^2 = x ∂_x + h_0   ,   
    V_-1^2 = x^2 ∂_x + 2 h_0 x   .

As in <cit.>, we express (<ref>) as

    W_h_0 (z_f ; z_i) =    ∑_n=0^∞( - 6/c)^n ∫_z_i^z_f dz_n ⋯∫_z_i^z_2 dz_1
    
       ×∑_s_j = 2^N  [ ∏_j=1^n c_s_j/N_s_j J^(s_j) (z_j) ] f_h_0^(s_n , … , s_1) (z_f;z_i;z_n , … , z_1)

with (X=x)

    f_h_0^(s_n , … , s_1) (z_f;z_i;z_n , … , z_1)
          = . [ e^(z_f - z_n) V^2_1  V^s_n_-s_n +1 e^ (z_n - z_f) V^2_1] ⋯[ e^(z_f - z_1) V^2_1  V^s_1_-s_1 +1 e^ (z_1 - z_f) V^2_1] e^(z_f - z_i) V^2_1 ⟨ X | hw⟩|_X=0

 in 1/c expansion.


Let us first evaluate the two point function of operator 𝒪_j for N=2  as

    ⟨𝒪_j (z) 𝒪_j (0) ⟩ = ⟨ W_h_0 (z;0) ⟩

with h_0 = - j by applying our Wilson line method.
At the leading order in 1/c, the two point function becomes

    . ⟨ W_h_0 (z;0) ⟩|_𝒪(c^0) =  1/z^ 2 h_0

as expected.
The contribution at the next order is

    . ⟨ W_h_0 (z;0) ⟩|_𝒪(c^-1)    = ( 6/c)^2 ∫_0^z dz_2 ∫_0^z_2 d z_1  f_h_0^(2,2) (z;0;z_2,z_1) ⟨ J^(2) (z_2) J^(2)(z_1) ⟩
        = 1/c z^2 h_0[ 6 (h_0 - 1)h_0/ϵ + 12 h_0 (h_0 - 1) log(z) + 2 h_0 (5 h_0 - 2) ]    ,

where we have used (<ref>).
We can remove the term proportional to 1/ϵ by changing the overall normalization of Wilson line operator. The shift of conformal weight can be read off from the term proportional to log (z) as in (<ref>), and the 1/c order term in (<ref>) is successfully reproduced.


We also examine a three point function using Wilson line operator as

    ⟨𝒪_j (z)𝒪_j (0)J^(2)(y)⟩ = ⟨ W_h_0 (z;0) J^(2)(y) ⟩

with the extra insertion of the spin 2 current. At the leading order in 1/c, we have

    lim_y → - ∞ |y|^4 . ⟨ W_h_0 (z;0) J(y) ⟩|_𝒪(c^0)
       = lim_y → - ∞ |y|^4 6/c∫_0^z dz_1 f^(2)_h_0 (z;0;z_1)⟨ J^(2) (z_1) J^(2) (y ) ⟩
    = h_0 z^2 . ⟨ W_h_0 (z;0) ⟩|_𝒪(c^0)

as in (<ref>). At the next leading order, there are two types of contributions as

    (6/c)^2  ∫_0^z dz_2 ∫_0^z_2 dz_1 f^(2,2)_h_0 (z;0;z_2 , z_1)⟨ J^(2) (z_2)  J^(2) (z_1) J^(2) (y ) ⟩  , 
       (6/c)^3  ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f^(2,2,2)_h_0 (z;0;z_3, z_2 , z_1)⟨ J^(2) (z_3) J^(2) (z_2)  J^(2) (z_1) J^(2) (y ) ⟩  .

We use the correlators

    ⟨ J^(2) (z_3) J^(2) (z_2) J^(2) (z_1) ⟩
     = c/(z_21  z_32  z_31)^2 - ϵ

for the three point function and

    ⟨ J^(s) (z_4) J^(s) (z_3) J^(s) (z_2) J^(s) (z_1) ⟩
    = B_s^2/(z_43  z_21)^2 s - 2 ϵ + B_s^2/(z_42  z_31)^2 s - 2 ϵ + B_s^2/(z_41  z_32)^2 s - 2 ϵ

with s=2 up to order c^2 for the four point function.
The sum of the integrals leads to

    lim_y → - ∞ |y|^4 - 2 ϵ⟨ W_h_0 (z;0) J^(2)(y) ⟩ 
    =[ h_0- 6 h_0/c( 1/ϵ + h_0 - 1/2) ]   z^2  ⟨ W_h_0 (z;0) ⟩ + ⋯  .

We remove the divergence by renormalizing the coupling c_2 in (<ref>) as (see (4.21) of <cit.>)

    c_2 = 1 + 1/c( 6/ϵ + 3 ) + 𝒪(c^-2)   .

Here the constant term is chosen such that the three point function becomes

    lim_y → - ∞ |y|^4 ⟨ W_h_0 (z;0) J^(2)(y) ⟩
    = h_jz^2 ⟨ W_h_0 (z;0) ⟩

with h_j in (<ref>) up to order 1/c. In other words, we chose the renormalization scheme to be consistent with the conformal Ward identity in the boundary theory.
Here, we remark that the renormalized coupling c_2 in (<ref>) is independent of h_0; thus,  the same Wilson line operator (<ref>) can be used to evaluate the network in (<ref>).


In <cit.>, we also examined the 1/c^2 order corrections of the two point function, and we reproduced the shift of conformal weight at the 1/c^2 order as in (<ref>). This is possible only after properly regularizing divergences and this fact gives a support for our renormalization prescription, see <cit.> for more details. The analysis here can be directly applied to the case with N=3 but involving scalar operators only in the (anti-)fundamental representation. Similarly for N=2, we showed in <cit.> that the divergences at the 1/c order in the two and three point functions of the form (<ref>) can be removed by renormalizing the overall factor of Wilson line operator and couplings c_2 and c_3 introduced as in (<ref>). With the renormalization at the 1/c order, we reproduced the shift of conformal weight of scalar operator up to the 1/c^2 order.




 §.§ Two and three point functions for N=3



In <cit.>, we restrict ourselves to operators only in the (anti-)fundamental representation. However, the fusion of this type of operators would yield another operator belonging to a representation other than the (anti-)fundamental one. Therefore, it is necessary to extend the previous analysis and deal with correlators involving general operators.
It is known that sl(N) generators acting on general states can be represented by using parameters x_a (a = 1,2, …, (N-1)N/2) with the number of the positive root.
For N=3, we introduce three parameters X=(x,y,w), which makes the expression (<ref>) with (<ref>) applicable to the general situation. The explicit expressions of states and generators are summarized in appendix <ref>.



Just like the N=2 case, we start from computing the two point function of operator 𝒪_(λ_1 , λ_2 ; 0) using the Wilson line as

    ⟨𝒪̅_(λ_1 , λ_2 ; 0) (z)  𝒪_(λ_1 , λ_2 ; 0) (0) ⟩ = ⟨  W_(λ_1 , λ_2) (z ; 0) ⟩  .

The expression of the Wilson line in 1/c expansions in (<ref>) with (<ref>) leads to

    .	⟨  W_(λ_1 , λ_2) (z) ⟩|_𝒪(c^0)
      =  z^2 λ_1 + 2 λ_2

with the normalization of (<ref>).
This is consistent with the leading order expression of conformal weight in (<ref>).
At the next leading order in 1/c, we have the contributions as (s=2,3)

    ( 6/c)^2  ∫_0^z dz_2 ∫_0^z_2 dz_1 f^(s,s)_(λ_1 ,λ_2) (z;0;z_2 ,z_1)
    	⟨ J^(s) (z_2) J^(s) (z_1) ⟩  ,

where f_(λ_1, λ_2)^(s,s) (z;0;z_2,z_1) are given by (<ref>) with (<ref>), (<ref>), and (<ref>). Moreover, we use the two point functions of currents  in (<ref>) with

    B_2 = c/2  ,    
    B_3 = - 5 c/6  ,

see (<ref>) with N=3.
The sum of the both contributions is

    ⟨ W_(λ_1 , λ_2) (z) ⟩ =     z^2 λ_1 + 2 λ_2  + 1/c z ^  2 λ_1 + 2 λ_2[
    	8 (λ_1^2+λ_1 (λ_2+3)  +λ_2 (λ_2+3)) /ϵ
       +1/15(240 (λ_1^2+λ_1 (λ_2+3)+λ_2 (λ_2+3)) log (z) . 
       . +197 λ_1^2+λ_1 (206 λ_2+213)+λ_2 (197 λ_2+213)) ]

up to the orders of ϵ^0 and 1/c. The z-independent overall factor can be set as 1 by changing the normalization of the Wilson line operator.
From the coefficient of  log(z), the shift of conformal dimension can be read off as

    - 1/2·1/c· 16 (λ_1^2+λ_1 (λ_2+3)+λ_2 (λ_2+3))    ,

which reproduces the 1/c order term in  (<ref>).



Next we move to the evaluation of Wilson line operator with the insertion of a higher spin current J^(s) with s=2 or s=3.
We would like to show that the three point functions can be reproduced as

    ⟨𝒪̅_(λ_1 , λ_2 ; 0) (z)  𝒪_(λ_1 , λ_2 ; 0) (0) J^(s) (y) ⟩ =   ⟨  W_(λ_1 , λ_2) (z ;0) J^(s) (y) ⟩  .

Let us define

    G_3^(s) (z) = lim_y → - ∞ |y|^2 s⟨ W_(λ_1,λ_2) (z) J^(s) (y)⟩

with s=2,3. We first examine the leading order in 1/c.
From the expression of the open Wilson line in (<ref>) with (<ref>), we find

    . G_3^(2) (z)  |_𝒪(c^0)
    = 6/c B_2 ∫_0^z dz_1 f^(2)_1 (z_1)
    =  - (λ_1 + λ_2) z^2 + 2 λ_1  + 2 λ_2  , 
       . G_3^(3) (z)  |_𝒪(c^0)
    = -  6/c B_3∫_0^z dz_1 f^(3)_1 (z_1)
    = 1/3  (λ_1 - λ_2) z^3 + 2 λ_1  + 2 λ_2

using (<ref>), (<ref>), and (<ref>).
There are no divergences from these tree level computations, which reproduce (<ref>) with (<ref>) and (<ref>) at this order.

Divergences arise from the next order in 1/c.
For G_3^(2) (z) in (<ref>) at this order, a type of contributions comes from the integrals as
[We take the limit G_3^(s) (z) |_𝒪(c^-1) = lim_y → - ∞ |y|^2 s - 2 ϵ⟨ W_(λ_1,λ_2) (z) J^(s) (y)⟩ |_𝒪(c^-1) for these computations.]

    ( 6/c)^2 c ∫_0^z dz_2 ∫_0^z_2  dz_1 f_(λ_1,λ_2)^(2,2) (z;0 ;z_2,z_1) 1/z_21^2 - ϵ  , 
       - ( 6/c)^2 5 c/2∫_0^z dz_2 ∫_0^z_2  dz_1 f_(λ_1,λ_2)^(3,3) (z;0 ; z_2,z_1) 1/z_21^4 -  ϵ  .

Here, we have used the three point functions (<ref>) and

    ⟨ J^(3) (z_3) J^(3) (z_2) J^(2) (z_1) ⟩ = - 5c/2/z_32^4 - ϵ z_21^2 - ϵz_31^2 - ϵ  ,

where the powers are obtained by shifting the conformal weights of J^(2) and J^(3) as 2 → 2 - ϵ and 3 → 3 - ϵ, respectively.
Another type is

    ( 6/c)^3 B_s B_2  ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(s,s,2) (z;0;z_3,z_2,z_1)1/z_32^2s - 2 ϵ  ,  
       ( 6/c)^3 B_s B_2  ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(2,s,s) (z;0;z_3,z_2,z_1)1/z_21^2s - 2 ϵ  , 
       ( 6/c)^3 B_s B_2  ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(s,2,s) (z;0;z_3,z_2,z_1)1/z_31^2s - 2 ϵ

with s=2,3.  Here the large c factorization of four point functions is used as (<ref>) and

    ⟨ J^(3) (z_4) J^(3) (z_3) J^(2) (z_2) J^(2) (z_1) ⟩
    = B_2 B_3/z_43 ^6 - 2ϵz_21^4 -2 ϵ + 𝒪(c)   .

Totally, we have

    z^-2 G_3^(2) (z)
    =[ - (λ_1 + λ_2)
    
          + 1/c( 36 (λ_1+λ_2) /ϵ-(8 λ_1^2+λ_1 (8 λ_2+11)+λ_2 (8 λ_2+11))  ) ]  ⟨ W_(λ_1 , λ_2) (z;0)⟩

up to orders ϵ^0 and 1/c.
There is  a divergent term proportional to 1/ϵ, which is removed by shifting the coupling c_2 introduced in (<ref>)  from its classical value c_2 = 1.
We choose the ϵ-independent part of c_2 as

    c_2 = 1 + 1/c( 36/ϵ + 13) + 𝒪 (c^-2)   ,

then we  reproduce the CFT result in (<ref>) with (<ref>) as

    z^-2 G_3^(2) (z)     = h(λ_1 , λ_2 ;0)⟨ W_(λ_1 , λ_2) (z)⟩

up to the order 1/c.
Note that the shift of coupling  c_2 in (<ref>) is the same as (5.17) of <cit.>, and it is independent of the representation (λ_1,λ_2).




We can similarly analyze G_3^(3)(z) in (<ref>).
At the 1/c order, contributions are

    ( 6/c)^2  5 c/2∫_0^z dz_2 ∫_0^z_2  dz_1 [ f_(λ_1,λ_2)^(2,3) (z;0,z_2,z_1) + f_(λ_1,λ_2)^(3,2) (z;0,z_2,z_1) ] 1/z_21^2 - ϵ  ,

and

    -( 6/c)^3 B_s B_3∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(s,s,3) (z;0,z_3,z_2,z_1)1/z_32^2s - 2 ϵ  ,  
       -( 6/c)^3 B_s B_3 ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(3,s,s) (z;0,z_3,z_2,z_1)1/z_21^2s - 2 ϵ  , 
       -( 6/c)^3 B_s B_3 ∫_0^z dz_3 ∫_0^z_3 dz_2 ∫_0^z_2 dz_1 f_(λ_1,λ_2)^(s,3,s) (z;0,z_3,z_2,z_1)1/z_31^2s - 2 ϵ

with s=2,3. Here we have used (<ref>), (<ref>), and (<ref>).
 The sum of them leads to

    z^-3 G_3^(3) (z)  =[  λ_1 - λ_2/3
         
    + 1/c( -12 (λ_1-λ_2) /ϵ + 1/5 (λ_1-λ_2) (40 λ_1+40 λ_2+33)  ) ]  ⟨ W_(λ_1 , λ_2) (z)⟩

up to orders ϵ^0 and 1/c. Changing  the coupling  c_3 in (<ref>) as (see (5.30) of <cit.>)

    c_3 =  1 + 1/c( 36/ϵ + 1) + 𝒪 (c^-2)   ,

 the above expression becomes

    z^-3 G_3^(3) (z)     = w(λ_1 , λ_2 ;0)⟨ W_(λ_1 , λ_2) (z;0)⟩

up to the 1/c order as in (<ref>) with (<ref>). Here, we  remark that the shift of c_3 is independent of λ_1,λ_2 as for c_2.

Adopting the regularization, we should be able to reproduce the shift of conformal weight up to 1/c^2 order just like the case with (λ_1,λ_2) = (1,0) in <cit.>. Here we do not repeat the computation since it is not necessary for the following analysis.




§ CONFORMAL BLOCKS FROM OPEN WILSON LINES



In this section, we evaluate four point conformal blocks of the form (<ref>)  by including quantum effects on the networks of open Wilson lines. 
For N=2, similar computations were done in <cit.>, but their formulation is  different from ours. 
In the next subsection we start from the identity blocks, which are computable by the products of two open Wilson lines. Since computations turn out to be quite similar to those in <cit.> for N=2, we focus on the N=3 case by applying the method developed in the previous section. In subsection <ref>, we examine general conformal blocks, where we have to really deal with the networks of open Wilson lines. 
We concentrate on the simpler case with N=2 and briefly discuss the N=3 extensions, see also appendix <ref>.



 §.§ Identity W_3 blocks



In order to express a conformal block for (<ref>) in terms of the Wilson line network as in (<ref>), we have to choose a proper singlet S from the product of representations 
Λ̅⊗Λ⊗Ξ̅⊗Ξ as explained in subsection <ref>.
For the identity block, we set S as the direct product of two singlets given by s_12∈Λ̅⊗Λ and s_34∈Ξ̅⊗Ξ. With the choice, the Wilson line network factorizes as

    G_4 (Λ_j | z_j) = ( ⟨ s_12| ∏_j=1,2 W_Λ_j (z_0;z_j) | hw⟩_j ) ( ⟨ s_34| ∏_l=3,4 W_Λ_l (z_0' ;z_l) | hw⟩_l )

with z_0 = z_0'. Setting z_0 = z_1 = ∞ and z_0' = z_3 = z with z_2 =1 and z_4=0, the Wilson line network becomes the product of two open Wilson lines. 
Therefore, the identity block should be computed as

    ⟨ G_4 (Λ_j | z_j)  ⟩ = ⟨ W_Λ (∞ ; 1) W_Ξ (z; 0) ⟩  ,

where the products of currents are evaluated by the correlators of W_N theory as before. 
In this subsection we explicitly compute (<ref>) for N=3 with Λ = (λ_1,λ_2) and Ξ = f, that is

    G^0_4(z)= ⟨ W_(λ_1 , λ_2) (∞ ; 1) W_f (z; 0) ⟩
    = lim_w →∞ w^2 h(λ_1,λ_2;0)⟨ W_(λ_1 , λ_2) (w ; 1) W_f (z; 0) ⟩  ,

by applying the method developed in the previous section.
We compare the result with the identity block from the CFT computation in (<ref>), see appendix A for some details. The 1/c expansion is given as (<ref>) with (<ref>), but now we set N=3 and m=1.


The leading order expression in 1/c is simply given by the product of two expectation values as

    . G_4^0(z)  |_𝒪(c^0)   =  . ⟨ W_(λ_1 , λ_2) (∞ ; 1) ⟩|_𝒪(c^0).
    	⟨ W_f (z; 0) ⟩ |_𝒪(c^0)  = z^2   .

This can be expressed in diagram (a) of figure <ref>.

Among higher order corrections in 1/c, there are contributions of the self-energy type as in other diagrams.
For contributions corresponding to diagrams (b)-(f), we replace .
⟨ W_f (z; 0) ⟩ |_𝒪(c^0) in (<ref>) the integrals of the forms

    ∫_0^z dz_n ⋯∫_0^z_2 dz_1 f^(s_n, … , s_1)_f (z;0;z_n , …,z_1)
     ⟨  J^(s_n)(z_n) ⋯ J^(s_1) (z_1) ⟩

with n=2,3,4 and s_j = 2,3. Here, the integrals with n=2 and n=3 correspond to diagrams (b) and (c). The integrals with n=4 include the four point functions of currents with several different terms as in (<ref>) and (<ref>), which are expressed by diagrams (d), (e), and (f).
The 1/c corrections to W_f (z; 0) up to order 1/c^2 have been evaluated in <cit.>, and
they can be summarized as the shift of conformal dimension from (<ref>) to (see (<ref>) for a general operator)

    1/z^2 h(f;0)  ,    h(f;0) = - 1 - 32/c - 1600/c^2 + 𝒪(c^-3)

after renormalizing the overall factor of the open Wilson line and the coupling constants c_s in (<ref>).
The other diagrams correspond to the similar corrections of another Wilson line W_(λ_1 , λ_2) (∞ ; 1), but they do not appear in the four point block G_4^0(z) (<ref>) since an operator is put at the infinity.

Non-trivial contributions start to enter from the next order in 1/c.
They correspond to diagram (a) of figure <ref>,

and the sum of them is

    H_1 (z) = H_1^(2) (z) + H_1^(3) (z)

with

    H_1^(s) (z) = ( 6/c)^2  ∫_1^∞ dz_2 ∫_0^z dz_1 	f^(s)_(λ_1, λ_2) (∞;1;z_2) f_f^(s) (z;0;z_1)
     ⟨ J^(s) (z_2) J^(s) (z_1) ⟩  .

With the two point functions of currents in (<ref>) with (<ref>), we evaluate the integrals as

    H_1^(2)(z) =  2 (λ_1 + λ_2)/c z^4 _2 F_1(2,2;4;z)   ,   
    	H_1^(3) (z)= - 2(λ_1 - λ_2)/15 c z^5 _2 F_1(3,3;6;z)   .

They correspond to the global blocks with spin 2 and 3 currents as the intermediate operators as they should be.
The coefficients are also given by the products of three point functions divided by a current-current two point function as

    H_1^(2) (z) =h_(f;0) h_(λ_1 , λ_2 ; 0)/B_2 z^4 _2 F_1 (2,2;4;z)   ,   
    	H_1^(3) (z)=  w_(f;0) w_(λ_1 , λ_2 ; 0)/B_3 z^5 _2 F_1 (3,3;6;z)

at the leading order in 1/c.
Here, the spin 2 and 3 charges were given by (<ref>) and (<ref>).


The other diagrams in figure <ref> yield loop corrections to interactions among an open Wilson line and an exchanged current. These diagrams can be evaluated precisely as for (<ref>) but with y replaced by z_1 or z_2. Therefore, after renormalizing the open Wilson lines and the coupling constants c_s, the net effects are summarized as the 1/c order terms in (<ref>) and (<ref>) as

    h_(f;0) h_(λ_1 , λ_2 ; 0) =  λ_1 + λ_2
    	+ 8 (7 λ_1 + λ_1^2 + 7 λ_2  + λ_2^2 + λ_1 λ_2)/c + 𝒪(c^-2)   , 
        w_(f;0) w_(λ_1 , λ_2 ; 0) = λ_1-λ_2/9 + 8 (15 λ_1^2+41 λ_1-15 λ_2^2-41 λ_2)/45 c+ 𝒪(c^-2)   .

Thus we have a type of 1/c^2 corrections as

    H_2^(2) (z)=   16 (7 λ_1 + λ_1^2 + 7 λ_2  + λ_2^2 + λ_1 λ_2)/c^2 z^4 F(2,2,4;z)   , 
       H_2^(3) (z)= - 16 (15 λ_1^2+41 λ_1-15 λ_2^2-41 λ_2)/75 c^2 z^5 F(3,3,6;z)

coming from diagrams (b)-(i) of figure <ref>.


At the order of 1/c^2, there are more interesting contributions with the exchange of two currents as in diagrams (a) and (b) of figure <ref> in addition to those already given.
When the two currents have the same spin, they come from the integrals

    H^(s,s)_2 (z)    = ( 6/c)^4  B_s^2 ∫_1^∞ dz_4 ∫_1^z_4 dz_3 ∫_0^z dz_2 ∫_0^z_2 dz_1  
       ×
    	f^(s,s)_(λ_1, λ_2) (∞;1;z_4,z_3)	f_f^(s,s) (z;0;z_2,z_1)
    	 (1/z_31^2s z_42^2s + 1/z_41^2s z_32^2s)

with s=2,3. The first and second terms are expressed by diagrams (a) and (b), respectively.
When the two currents have the different spins, the contributions corresponding to diagrams (a) and (b) are given by the integrals

    H^(s_1,s_2 )_2,1 (z)    = ( 6/c)^4  B_2 B_3 ∫_1^∞ dz_4 ∫_1^z_4 dz_3 ∫_0^z dz_2 ∫_0^z_2 dz_1  
       ×
    	f^(s_2,s_1)_(λ_1, λ_2) (∞;1;z_4,z_3) 
    	f_f^(s_2,s_1) (z;0;z_2,z_1)
    1/z_31^2s_1 z_42^2s_2  , 
    
    	H^(s_1,s_2 )_2,2 (z)    =  ( 6/c)^4  B_2 B_3 ∫_1^∞ dz_4 ∫_1^z_4 dz_3 ∫_0^z dz_2 ∫_0^z_2 dz_1  
       ×
    	f^(s_1,s_2)_(λ_1, λ_2) (∞;1;z_4,z_3) 
    	f_f^(s_2,s_1) (z;0;z_2,z_1)
    1/z_41^2s_1 z_32^2s_2

with (s_1,s_2) = (2,3) or (3,2).

Summing over this type of contributions, we find

    H_2^2ex (z) 
       = 8/5[20 log ^2(1-z) (3 (λ_1^2+λ_1 (4 λ_2+6)+(λ_2-3) λ_2)+z^2 (2 λ_1+λ_2) (2 λ_1+λ_2+9) 
        -6 λ_1 z (λ_1+2 λ_2+6))+3 z (10 λ_1^2 (31 z-30)+λ_1 (160 λ_2 z+519 z-318)  
        +λ_2 (10 λ_2 (z+30)+201 z+318))-6 log (1-z) (10 λ_1^2 (8 (z-3) z+15) 
        +λ_1 (2 z (30 λ_2 (z-2)+7 z+3)-21)+λ_2 (10 λ_2 (z (z+6)-15)+(z-36) z+21) )
        +360 Li_2(z) (λ_1 (2 (z-3) z+3)+λ_2 (z^2-3) ) ]

with Li_2 (z) = ∑_n=1^∞ z^n/n^2.
The total contribution at the 1/c^2 order other than the self-energy type is thus

    H_2 (z)= H_2^(2) (z) + H_2^(3) (z) + H_2^2ex (z)

with (<ref>) and (<ref>).

We confirm the bulk computation by showing that the above result reproduces the CFT one. The identity block in (<ref>) is expanded in 1/c as (<ref>) with (<ref>), and we set N=3, m=1 and identify (λ_1,λ_2)=(n,0) for the comparison.
We can see that  the 1/c order term in (<ref>) is the same as (<ref>) with (<ref>).
Furthermore. we can check that the 1/c^2 order expression of (<ref>) with (<ref>) is reproduced by (<ref>) at each order in z expansion.


Before ending this subsection, we would like to mention the relation to the analysis in <cit.> for N=2 with a different regularization. The regularization prescription in <cit.> is to adopt the normal ordered product for currents coming from the same Wilson line operator. Moreover, they used the exact conformal weight h_j from the beginning instead of the quantum number j for the finite representation of sl(2). Using the normal ordered prescription is equivalent to neglecting contributions coming from loop diagrams in figures <ref> and <ref>, see also <cit.>. As we explained above, these contributions only shift the conformal weight for N=2; therefore, the effect is already included in their treatment. The contributions represented by diagrams in figure <ref> are the same in the both prescriptions, see figure 5 of <cit.>. Therefore, we can see that the both ways of computation should give the same result for N=2.



 §.§ General conformal blocks



In this subsection, we study the conformal blocks with the exchange of the general operator from the networks of open Wilson lines (<ref>) by including quantum effects.
For the general blocks, it turns out to be convenient to use the X-basis as

    G_n(Λ_j | z_j) =[  ∏_j = 1^n ∫ d X_j   _j⟨lw | X_j ⟩  W_Λ_j (z_j  ; z_0 ; X_j )   ⟨ X_j|  ] | S ⟩

in its conjugated form. Here the generators V_m^s(X_j) are now written in terms of X_j 
as explicitly indicated in the arguments of the Wilson line operator.
The condition for the singlet |S ⟩ becomes

    ∑_j=1^n  V_m^s (X_j)  [ ∏_l=1^n ⟨ X_l | ] | S ⟩  = 0

for all s and m. 
In general there are several solutions, and a solution corresponds to a conformal block.
Applying _j⟨lw | X_j ⟩  = δ (X_j) as in (<ref>) or (<ref>), the expression is simplified as

    G_n(Λ_j | z_j) = . [  ∏_j = 1^n   W_Λ_j (z_j  ; z_0 ; X_j )   ⟨ X_j|  ] | S ⟩|_X_j = 0  .

The quantum effects are evaluated by using the correlators of higher spin currents inserted as before.

In the following we mainly focus on the N=2 case and make some comments on the N=3 case at the end of this subsection.
Before going into the details on three and four point Virasoro blocks, let us mention how the formalism explained above reproduces the previous analysis on the two point function for N=2.
We examine G_2 (j,j|z,0) in (<ref>) with X_i = x_i.
The vertex is given by the unique solution to the singlet condition (<ref>) as (see, e.g., <cit.>)

    ⟨ x_1 |   ⟨ x_2|| S ⟩ = v_2(j , j | x_1 , x_2)  ≡   x_12^2 j

up to the overall factor. 
At the leading order in 1/c, we have

    . ⟨ G_2 (j , j | z , 0) ⟩|_𝒪(c^0) = .  e ^(z - z_0)V_1^2 (x_1)  e ^ - z_0V_1^2 (x_2) 
    v_2 (j , j | x_1 , x_2) |_x_i = 0
    = z^2 j

as in (<ref>). For the 1/c corrections, it is convenient to set z_0 = 0 (or z_0 =z), which reduces the number of diagrams, see figure <ref>.

With z_0 = 0, the expression in (<ref>) reduces to the expectation value of (<ref>) with (<ref>) and (z_f,z_i) = (z,0). We have also evaluated the 1/c corrections to the two point function with keeping z_0 generic and checked that the final result is the same.




  §.§.§ Three point Virasoro block


As a simple but non-trivial example, we study three point conformal block for N=2. The three point block is fixed by the conformal symmetry as

    ⟨𝒪_j_1 (z_1) 𝒪_j_2 (z_2) 𝒪_j_3 (z_3) ⟩∝1/z_12^h_j_1 + h_j_2 - h_j_3 z_13^h_j_1 + h_j_3 - h_j_2 z_32^h_j_3 + h_j_2 - h_j_1

with (see (<ref>))

    h_j_i = - j_i +  1/c h_i^(1) + 𝒪(c^-2)   ,    h_i^(1) = - 6 j_i (j_i + 1)   .


We evaluate G_3 (j_i | z_i) in  (<ref>) with X_i = x_i.
The vertex is uniquely fixed by the singlet condition (<ref>) as  (see, e.g., <cit.>)

    [ ∏_i=1^3 ⟨ x_i | ] | S ⟩ = v_3(j_i | x_i)  ≡   x_12^j_1 + j_2 - j_3 x_13^j_1 + j_3 - j_2 x_32^j_3 + j_2 - j_1

up to the overall factor.
At the leading order in 1/c, we find

    . ⟨ G_3 (j_i | z_i) ⟩|_𝒪(c^0) = . [ ∏_j=1^3 e ^z_j0V_1^2 (x_j) ]
    v_3(j_i | x_i) |_x_i = 0
    = z_12^j_1 + j_2 - j_3 z_13^j_1 + j_3 - j_2 z_32^j_3 + j_2 - j_1  .

The expression reproduces the three point function at the leading order in 1/c.
Here we simply replace x_i by z_i,
[Precisely speaking, x_i are replaced by z_i0  (≡ z_i -z_0) with z_0 as the position of vertex for the Wilson line network. Since the results do not depend on z_0, we can set z_0 = 0.]
and this is actually as the same as the prescription to obtain the large c limit of three point Virasoro block in <cit.>.

We move to the next leading order in 1/c.
When evaluating the expectation value of G_3 (j_i | z_i), the correlators of J^(2) (z) for Virasoro algebra are used, and divergences are removed as before.
In order to make the computation simpler, we set (z_1,z_2,z_3) = (z,0,1).
Moreover, using z_0 = z_3 = 1,
[Similarly to the two point function, we have checked that the final result in (<ref>) does not depend on z_0 even for generic z_0. For more complicated correlators, the computations would be quite messy without setting z_0 as a specific value.]
we consider the product of two Wilson lines as in figure <ref>.

With this setup, we evaluate the integrals corresponding to the three diagrams in the right hand side.
Defining

    t_1 ( z ; x_1 , x_2 , x_3) =  ( 6/c)^2  ∫_1^z dz_2 ∫_1^z_2 d z_1
    ∏_j=1^2 [ e^(z - z_j) V_1^2 (x_1) V_- 1^2 (x_1) e^(z_j - z) V_1^2  (x_1)]
    
         ×
    e^(z-1) V_1^2 (x_1) e^-V_1^2 (x_2) v_3(j_i | x_i)
    ⟨ J^(2)(z_2) J^(2)(z_1) ⟩  ,  
       t_2  ( z ; x_1 , x_2 , x_3)  =  ( 6/c)^2 ∫_1^0 dz_2 ∫_1^z_2 d z_1
    ∏_j=1^2 [ e^ - z_j V_1^2 (x_2) V_- 1^2 (x_2) e^ z_j V_1^2 (x_2)]         
         ×
    e^(z-1) V_1^2 (x_1) e^-V_1^2 (x_2) v_3(j_i | x_i)
    ⟨ J^(2)(z_2) J^(2)(z_1) ⟩  , 
        t_3  (z ; x_1 , x_2 , x_3) = ( 6/c)^2 ∫_1^0 dz_2 ∫_1^z d z_1
    [ e^ - z_2 V_1^(2) (x_2) V_- 1^(s)  (x_2) e^ z_2 V_1^(2)  (x_2)]  
         ×[ e^(z - z_1) V_1^2 (x_1) V_- 1^2 (x_1) e^(z_1 - z) V_1^2 (x_1)]  e^(z-1) V_1^2 (x_1) e^-V_1^2 (x_2) v_3(j_i | x_i)
    ⟨ J^(2)(z_2) J^(2)(z_1) ⟩  ,

the integrals corresponding to diagrams (a), (b), and (c) are
t_a (z ; 0,0,0) with a=1,2,3, respectively.
The 1/c order correction to the three point conformal block is given by the sum of them as

    . ⟨ G_3 (j_i | z_i) ⟩|_𝒪(c^-1) =    ∑_a=1^3 t_a (z ; 0,0,0) =
    - 1/c z^j_1+j_2-j_3  (z-1)^j_1+j_3-j_2
       ×[ (h^(1)_1 + h^(1)_2 - h^(1)_3) log(z) + (h^(1)_1 + h^(1)_3 - h^(1)_2) log (z-1) ]

with h_i^(1) in (<ref>) up to the order ϵ^0 term. Here we have ignored the term proportional to the tree level expression, which is interpreted as a correction to the overall factor of three point function.
The result reproduces the 1/c order correction of three point block in (<ref>) with (<ref>) by setting (z_1,z_2,z_3) = (z,0,1).




  §.§.§ Four point Virasoro block



We move to the Virasoro block of operator 𝒪_p for the four point function (<ref>). For this correlator, we may set (z_1,z_2,z_3,z_4) = (z,0,∞,1) and (j_1,j_2,j_3,j_4) = (j,j,q,q).
We compute G_4(j_i | z_i) in (<ref>) with X_i = x_i.
A solution to the singlet condition (<ref>) is given by <cit.>

    [ ∏_i=1^4 ⟨ x_i | ] | S ⟩ = v_p(j_i | x_i)   ,

where we have defined 

    v_p (j_i | x_i) ≡𝒞∫ d x   v_3 (j ,j , p | x_1 ,x_2 ,x) v_3 (q,q , -1 -p|  x_3 ,x_4 , x)   .

With the integral contour as x ∈ (0 , ∞) and the overall factor as

    𝒞 = (-1)^p Γ ( -2 p)/(Γ (-p))^2   ,

the leading order expression in 1/c becomes

    . ⟨ G_4(j_i |  z_i) ⟩|_𝒪(c^0) = . [ ∏_j=1^4 e ^z_j0V_1^2 (x_j) ]
    v_p (j_i | x_i) |_x_i = 0 = z^2j - p g_p (z)   ,

where z^2j - p g_p (z) is the  global block of operator 𝒪_p as

    z^2j - p g_p (z)  ≡ z^2j - p_2 F_1 (-p,-p;-2p;z)   .

Just like the  three point block, this is essentially the way to obtain the large c limit of the four point block in <cit.>.



For the next leading order in 1/c, we evaluate the network of open Wilson lines with z_0 = z_4 = 1. With this setting, the number of diagrams we have to consider is reduced as in figure <ref>.

We choose the normalization of conformal block as

    ⟨ G_4(j_i | z_i) ⟩  = z^2j - p (1 + 𝒪(z) )   ,

and hence we subtract the terms proportional to the global block (<ref>) such that there is no z^2j -p term at the 1/c order.
With t_a (z;x_1,x_2,x_3) defined in (<ref>),
the integrals corresponding to diagrams (a), (b), and (e) are written as

    𝒞∫_0^∞ d x    t_a ( z ; 0,0,x) x^ - 1 - p

with a=1,2,3, respectively.
For other diagrams, we define the functions as

    u_1 (z ; z' ;  x_1 , x_2 , x_3) =   [ e^(z - z') V_1^2 (x_1) V_- 1^2 (x_1) e^(z' - z) V_1^2 (x_1)]
    e^(z - 1) V_1^2 (x_1) e^ - V_1^2 (x_2) v_3 (j_i | x_i)  , 
    
    u_2 (z ; z' ; x_1 , x_2 , x_3) =   [  e^ - z' V_1^2 (x_1) V_- 1^2 (x_2) e^ z' V_1^2 (x_1)] e^(z - 1) V_1^2 (x_1) e^ - V_1^2 (x_2) v_3 (j_i | x_i)

with j_1 = j_2 = j, j_3 = p, and

    u_3 (z' ;  x_1 , x_2 , x_3)   =    lim_w →∞ w^-2 q[e^(w - z') V_1^2 (x_1) V_- 1^2 (x_1) e^(z' - w ) V_1^2  (x_1)]
    e^(w - 1) V_1^2 (x_1) v_3 (j_i | x_i)

with j_1 = j_2 = q , j_3 = - 1 - p.
Then the integrals corresponding to diagrams (d) and (f) are

    𝒞(6/c)^2  ∫_1^z dz_2 ∫_1^∞ dz_1 ∫_0^∞ dx   u_1(z ;z_2 ; 0,0,x) u_3 (z_1 ;0,0,x)  ⟨  J^(2) (z_1) J^(2) (z_2) ⟩  , 
       𝒞(6/c)^2∫_1^0 dz_2 ∫_1^∞ dz_1 ∫_0^∞ d x   u_2(z ;z_2 ; 0,0,x) u_3 (z_1;0,0,x)  ⟨ J^(2) (z_1) J^(2) (z_2) ⟩  ,

respectively.
The sum of all contributions is computed as

    . ⟨ G_4 (j_i | z_i) ⟩|_𝒪(c^-1) = 
     6  z^2 j-p-1/c{_2F_1(-p,-p;-2 p;z)  [ z log (z) (2 j (j+1)- p (p+1))
     
            + log (1-z) (2 j q (z-2)+p^2 (z-1)+p (z-1) )
    +z (2 j-p) (p-2 q)]    
             -  _2F_1(1-p,-p;-2 p;z) log (1-z)  p (z-1) (2 j+2 q+1)  }

up to the terms proportional to (<ref>).
Here we neglected the integral corresponding to diagram (c) since it can be shown to give only terms proportional to (<ref>) or h_p log(w) with w →∞.


Let us compare the 1/c order expression with the known result in (<ref>).
First we examine the log (z) term in (<ref>).
This type of contribution comes from the shifts of conformal weights as

    z^ - 2 h_j  + h_p_2 F_1 (h_p ,h_p ;2 h_p;z) 
          =  z^2j - p g_p (z)
    + 1/c (- 2 h^(1)_j + h^(1)_p) log(z)   z^2j - p g_p (z)+ 1/c z^2j - p r_p (z) + 𝒪(c^-2)

with

    r_p (z) = - 12 p (p+1)  ∑ _n=0^∞[ (-p)_n]^2/ (-2 p)_nz^n /n!∑ _k=1^n (1/k-p-1-1/k-2 p-1)   .

We can see that the term proportional to log (z) successfully reproduces the corresponding term in (<ref>).
There is another type of contribution summarized as

    jq/c f_a (-p,z) - j + q/cf_b (-p,z) + 1/c f_b (-p,z)   .

We read off the functions f_a,f_b,f_c from (<ref>) as

    f_a (-p,z) = 12 z^2 j-p-1  _2F_1(-p,-p;-2 p;z) (log (1-z)  (z-2)- 2 z )   , 
       f_b (-p,z) = - 12 p z^2 j-p-1 ( z   _2F_1(-p,-p;-2 p;z)- (z-1) log (1-z)   _2F_1(1-p,-p;-2 p;z))   , 
       f_c(-p,z) = 6 z^2j -p-1[   _2 F_1(-p,-p;-2 p;z) ((p^2 (z-1)+p (z-1)) log (1-z)-p^2 z) 
                 -p (z-1) log (1-z)   _2 F_1(1-p,-p;-2 p;z)]-z^2j - p r_p (z)   .

The functions f_a ,f_b reproduce those in <cit.>.
For the other function f_c, the terms proportional to z^2j - p + l (l=2,3,4,5) were obtained in <cit.>, and the z-expansion of the above result correctly reproduces them.
In other words, our method enables us to obtain the all-order expression in z at the next non-trivial order in 1/c, which was not available before.

One may wonder how this becomes possible. In <cit.>, the authors evaluated the operator product expansion (OPE) blocks in the sense of <cit.> in terms of open Wilson lines and then computed the four point Virasoro blocks using the expression of the OPE blocks.
The procedure requires the integration over z_5, z_6 for the two point function of the intermediate state as ⟨𝒪_p (z_5) 𝒪_p (z_6) ⟩, while
we need to integrate over only one parameter x except for the positions z_i of the currents inserted.
We think that this is a main reason for the simplicity.



  §.§.§ General W_3 blocks




So far we have analyzed general Virasoro blocks, but we also would like to make some comments on general W_3 blocks.
We are interested in the expectation value of G_n (Λ_j|z_j) in (<ref>), where
we can obtain the vertex of open Wilson lines by solving the singlet condition in (<ref>).
Solutions to the condition were obtained in <cit.> for n=2,3,4,
which may be denoted as

    [ ∏_j=1^n ⟨ X_j | ] | S ⟩  =  E_n (Λ_j | X_j)

for the n-point function.
For n=3, the representation Λ = (λ_1 , λ_2) of an external operator is restricted to be degenerated as λ_1 =0 or λ_2 =0. A solution for n=4 is written as an integral of the product of two n=3 solutions, so there are also similar restrictions on the representations of external operators.

We first indicate that our formulation reduces to the one in <cit.> at the leading order in 1/c.
At this order, the expression in (<ref>) reduces to

    . ⟨ G_3 (Λ_j | z_j) ⟩|_𝒪(c^0) = . [ ∏_j=1^3 e ^z_jV_1^2 (X_j) ]
     E_n (Λ_j | X_j)  |_X_j = 0  ,

where we have set z_0 =0 by using the independence of z_0.
Utilizing the formula (<ref>), we can see that the leading order expression is obtained from 
E_n (Λ_j | X_j) by replacing X_j = (x_j , y_j ,w_j) with (z_j ,z_j , z_j^2/2).
Just like the N=2 case, the recipe to obtain the large c limit of the W_3 block is the same as the one in <cit.>.
As a result of this fact, we can conclude that our formulation with Wilson lines reproduces the large c limit of W_3 blocks as obtained in <cit.>, see also <cit.>.
Similar analysis with Wilson lines was done in <cit.> with a different technique for algebraic calculations. Our method with the X-basis makes computations easier for  generic representations as seen above, but their method seems to be convenient for a simple representation, such as, Λ = f. In appendix <ref>, we compute the 1/c correction of a simple non-identity W_3 block from the Wilson line network (<ref>) by utilizing their analysis.



We can also show that the analysis of two point function reduces to the previous one in subsection <ref>. For n=2, the solution to the singlet condition (<ref>) was obtained in <cit.> as

    E_2 (Λ , Λ̅ | X_1 , X_2) =2^λ_1 + λ_2ρ_12^λ_1ρ_21^λ_2  ,   ρ_ij = y_i (x_i - x_j) - (w_i - w_j)   .

For z_0 =z_2, the expression in (<ref>) becomes

    G_2 (Λ , Λ̅ | z_1 , z_2) =
    . W_Λ (z_1 ; z_2 ; X_1) ⟨ X_1 | hw⟩|_X_1 = 0

as for (<ref>) with (<ref>). Here, we have used

    . ρ_12^λ_1ρ_21^λ_2|_X_2 = 0
    = ⟨ X_1 | hw⟩  ,

where ⟨ X | hw⟩ was defined in (<ref>) with Λ = (λ_1 , λ_2).
Similarly, for z_0 = z_1, we obtain 

    G_2 (Λ , Λ̅ | z_1 , z_2) =
    . W_Λ̅ (z_2 ; z_1 ; X_2) ⟨ X_2 | hw⟩|_X_2 = 0  ,

but now ⟨ X | hw⟩ is with  Λ̅ = (λ_2 , λ_1).




§ HEAVY-LIGHT CORRELATOR



The W_N minimal model includes light operators of order c^0 and heavy operators of order c as explained in section <ref>.
Heavy operators correspond to non-trivial geometries,
and correlators with heavy operators are important for examining their quantum effects. 
In this section, we examine a heavy-light correlator for the simplest case with N=2,
and in section <ref>, we make some comments on the general cases with N ≥ 3.


For the analysis in this section, it is convenient to adopt the usual convention for the label of primary operators by two positive integers (r,s), see <cit.> for a review.
In section <ref>, we have used the numbers of boxes in Young diagrams as (n_1^+ ; n_1^-) and the relation is r = n_1^+ +1 , s =n_1^- + 1. The conformal weight of primary operator 𝒪_r,s is given by (<ref>) as

    h_r,s   = (r (k+3) - s (k+2) )^2 - 1/4 (k+2) (k+3)
       = - (s^2 - 1 ) c/24 + 1/24 (- 12 rs + 13 s^2  - 1) + 𝒪 (c^-1)   ,

which is of order c for n_1^- = s - 1 ≠ 0.
We study the four point correlator in (<ref>) with m=1, n = s -1 for N=2, 
the expression with finite k of which can be found in (<ref>). 

It is also useful to work on the cylindrical coordinate w, which is related to the planar one as z = exp (i w). In particular, the conformal weight is shifted from h_r,s in (<ref>) to h_r,s - c/24.
The expression of the four point function in (<ref>) becomes
[We have changed the phase factor for convenience, but it does not matter when we consider the product with the anti-holomorphic sector.]

    ⟨𝒪_1,s (- i ∞) 𝒪_2,1 (0) 𝒪_2,1 (w) 𝒪_1,s (i ∞) ⟩  =    (2 sins w/2) (- 2 i sinw/2)^9/c /s
                 - 6 i /c
    	e^- i s w/2∑_l=1^s-1(-1)^l (1-e^i w)^l+1Γ (s)/Γ (s- l) Γ (l+2)( ∑_j=1^l 1/j -  2 l/l+1) + 𝒪 (c^-2)

in the 1/c expansion.
In the rest of this section, we shall reproduce it from the bulk viewpoint.



 §.§ Conical spaces in AdS_3 gravity


As explained in section <ref>, 𝒪_1,s and 𝒪_2,1 are supposed to be dual to a conical geometry and a bulk scalar field, respectively. For the heavy-light correlator in (<ref>), we evaluate the expectation value of the open Wilson line from 0 to w in the background of a conical defect sitting at the center of AdS_3.
In this subsection, we review the analysis on the quantum effects of conical spaces in <cit.>, which will be used in the next subsection for the bulk computation of the heavy-light correlator.


In the Chern-Simons formulation, a geometry is realized as a configuration of the gauge field.
We express the gauge field in the sl(2) Chern-Simons theory with the cylindrical coordinate as

    a (w) = V_1^2 - 6/c J^(2) (w) V^2_-1

instead of (<ref>).
[The relative sign in front of J^(2) (and the shift of conformal weight) can be explained from the transformation of the energy momentum tensor under the change of coordinates as (<ref>) with (<ref>).]
The conical defect at the center of AdS_3 corresponds to the classical configuration  as <cit.> (s = 1,2,3,…)

    L_0 = -  s^2 c /24  ,    L_n ≠ 0 = 0

with the mode expansion of J^(2)(w) as

    J^(2)(w) = ∑_n L_n e^-i n w  .

In particular, the AdS vacuum is given by the case with s = 1.


Quantization of the conical spaces is analyzed in <cit.> utilizing the coadjoint orbits of the Virasoro algebra <cit.>.
The conical defect at the center of AdS_3 is represented by operators inserted at w = i ∞ and w = - i ∞, and the corresponding states are denoted as | 0 ⟩_s and _s ⟨ 0 |.
The states | 0 ⟩_s and _s ⟨ 0 | are defined as the Fock vacua of quantum operators A_m as

    A_m | 0 ⟩_s = 0   ,   _s ⟨ 0 |A_-m  = 0   ,    _s ⟨ 0 |0 ⟩_s = 1

with m > 0. Here A_m satisfies

    [A_m , A_n] = δ_m+n/m (m^2 - s^2)  .

The asymptotic symmetry is given by the Virasoro algebra generated by L_n, which satisfies

    [L_m , L_n] = (m - n) L_m+n + c/12 m^3 δ_m+n  .

These generators are written in terms of A_m as

    L_0 = -  s^2 c /24 + n_0 + ∑_m =1^∞ m^2 (m^2 - s^2) A_-m A_m + 𝒪 (c^-1/2 )    ,

 see <cit.> for the others.
Requiring that  the generators satisfy (<ref>), the number n_0 is determined as 

    n_0 = 1/24 (s -1) (1 + 13 s)   .

In particular, we have

    L_m | 0 ⟩_s = 0   ,   _s ⟨ 0 |L_-m  = 0   ,    _s ⟨ 0 | L_0|0 ⟩_s = - s^2 c/24 + n_0 + 𝒪 (c^-1/2)

with m > 0. The L_0 eigenvalue is the same as h_1,s - c/24 in (<ref>) up to the next non-trivial order in 1/c, which confirms the  identification as | 0 ⟩_s = 𝒪_1,s(i ∞)|0 ⟩ and _s ⟨ 0 | = ⟨ 0 | 𝒪_1,s ( - i ∞).



 §.§ Open Wilson line in conical space



We consider the expectation value of the open Wilson line in the conical geometry with label s.
As explained in subsection <ref>, we compute

    W_s (w)= . P exp[ ∫_0^w dw' ( V_1^2 - 6/c J^(2)(w') V_-1^2) ]  x |_x=0  ,

where the background configuration for J^(2) (w') is set as in (<ref>).
The generators V^2_n are given by (<ref>) with h_0 = -1/2.
Here, we set the coupling in (<ref>) as c_2=1, which is enough for the analysis in this subsection.


We evaluate its expectation value by expanding in 1/c.
Since J^(2)(w') is now of order c, we define a new operator,

    Ĵ^(2)(w') ≡ J^(2)(w') + s^2 c/24  ,

which is of order c^1/2 as in (<ref>) below. 
As before, we find

    d/d w[ e^- (V^2_1 + s^2/4 V^2_-1) w W_s (w) ]
    	
          = - 6/cĴ^(2) (w)   (V^2_-1)_s ( - w)
    	[ e^- (V^2_1 + s^2/4 V^2_-1) w W_s (w) ]

with

    (V_-1^2)_s ( w) ≡ e^ (V_1^2 + s^2/4 V_-1^2) w  V_-1^2 e^- (V_1^2 + s^2/4 V_-1^2) w  .

Using the equation, the 1/c expansion is given by

    W_h_0 (w)
    	= ∑_n=0^∞( - 6/c)^n ∫_0^w d w_n ⋯∫_0^w_2 dw_1
    	[ ∏_j=1^n Ĵ ^(2) (w_j)  ] f_s (w_n , … , w_1)

with

    f_s (w_n , … , w_1) =   .(V^2_-1)_s (w_*n) ⋯  (V^2_-1)_s (w_*1) e^ (V^2_1 + s^2/4 V^2_-1) w  x |_x=0  ,    w_*i≡ w - w_i  .

From (see (7.6) and (7.7) of <cit.>)

    e^(V^2_1 + s^2/4 V^2_-1) w = e^c_1 (w) V^2_1 [c_0 (w)]^2 V^2_0 e^ c_-1 (w) V^2_-1

with

    c_1 (w) = 2/stans w/2  ,   
    	c_0 (w) = coss w/2  ,   
    	c_-1 (w) = s/2tans w/2  ,

we find

    e^(V^2_1 + s^2/4 V^2_-1) w x
    	= c_0 (w) ( x + c_1 (w)  )

and

    (V^2_-1)_s ( w)  = c_0(w)^2 [ V^2_-1 + 2 c_1 (w)  V^2_0  + c_1 (w)^2 V^2_1]   .

In particular, the leading order expression in 1/c is obtained as

    . W_s (w)  |_𝒪 (c^0)= 2/ssins w/2  ,

which reproduces (<ref>).


 At the next leading order in 1/c, there are two types of contributions with one and two insertions of Ĵ^(2).
With one insertion, we compute

    H_s^(1) (w)    = -	6/c∫_0^w  dw_1    f_s (w_1) _s ⟨ 0 |Ĵ^(2)(w_1) | 0 ⟩_s 
        = 12 n_0/c [ 2sin(s w/2) - s w cos(s w/2) /s^3]   ,

where we have used

    _s ⟨ 0 | Ĵ^(2)(w_1) | 0 ⟩_s =  n_0  + 𝒪 (c^-1)   ,

see (<ref>).
Here we do not need to worry about the regularization issue.
In the following, we examine the case of two insertions with special care on the issue.



With the two insertions of Ĵ^(2), the integral is

    (-6/c)^2 ∫_0^w  dw_2  ∫_0^w_2  dw_1 f_s (w_2,w_1)  _s ⟨ 0 | Ĵ^(2) (w_2)  Ĵ^(2)(w_1) | 0 ⟩_s

with 

    f_s (w_2,w_1) =  8/s^3sin(s w_1/2) sin(s (w- w_2)/2) sin(  s (w_2 -w_1)/2)   .

Moreover, we find

    .  _s ⟨ 0 | Ĵ^(2)(w_1) Ĵ^(2) (w_2) | 0 ⟩_s |_𝒪(c)=  ∑_m,n > 0 e^- i(m w_1 - n w_2).
    _s ⟨ 0 | [ L_m  , L_-n ]| 0 ⟩_s |_𝒪(c)
          = c/12∑_m > 0 e^- i m (w_1 - w_2) (m^3 - s^2 m)
    = c/16( 1/2 sin ^4  ( w_1 - w_2/2) +  s^2 -1/3 sin ^2 ( w_1 - w_2/2))   .

However, the integration over w_1,w_2 diverges in (<ref>), and we have to properly regularize it.

In order to make the expression simpler, we introduce new parameters as

    z = e^ i w  ,    z_1 = e^i w_1  ,    z_2 = e^i w_2  ,   
    	y = e^ i s w  ,    y_1 = e^i s w_1  ,    y_2 = e^i s w_2  .

Then, the expression in (<ref>) can be rewritten as

    . _s ⟨ 0 | Ĵ^(2) (w_1) Ĵ^(2) (w_2) | 0 ⟩_s |_𝒪(c)=
    c/2z_1^2 z_2^2/(z_1 - z_2)^4 - c/12(s^2 -1) z_1 z_2/(z_1 -z_2)^2  .

The integral (<ref>) diverges due to (y_1 - y_2)^-4 singularity in (<ref>) as discussed in <cit.>, so we separate the singular part in (<ref>) as

    . _s ⟨ 0 | Ĵ^(2) (w_1) Ĵ^(2)(w_2) | 0 ⟩_s |_𝒪(c) =
    	g_1[w_1 , w_2] + g_2 [w_1 ,w_2]   , 
       1/s^4 (z_1 z_2)^2s g_1 [w_1 , w_2] =  c (z_1 z_2 )^2-2 s/2 s^4 (z_2 - z_1)^4
    	-c (s^2-1) (z_1 z_2)^1-2 s/12 s^4 (z_2 - z_1)^2 -c /2 (y_1 - y_2)^4  , 
       1/s^4 (z_1 z_2)^2s g_2 [w_1 , w_2] =
    	c /2 (y_1 - y_2)^4   .

The integral with g_1[w_1,w_2] was performed in <cit.>, which yields a finite result.
The integral with g_2[w_1 ,w_2] diverges so we need to regularize it in a proper way.


The integral with g_1[w_1,w_2] can be written as

    H^(2)_s (w)
       =   6^2 i z^-s/2/c s^3∫_1^z d z_2 ∫_1^z_2 d z_1 z_1^s-1z_2^s-1(1-z_1^s) (z_2^s-z^s) (z_1^s-z_2^s) 
       ×((z_1 z_2)^2-2 s/2 (z_2-z_1)^4 -(s^2-1) (z_1 z_2)^1-2 s/12 (z_2-z_1)^2 -s^4/2 (z_1^s-z_2^s)^4)

by using the explicit form of f_s(w_2,w_1) in (<ref>).
The expression with general complex s after the integration over z_1,z_2 was obtained in <cit.>. However, several terms diverge for positive integer s, and we should take care of cancellations among them.
Instead of doing so, we evaluate the integral directly with the explicit values of s.
As for illustration, we write down the expressions for s=2,3,4 as

    c s^3/6^2 i z^-s/2 H_2^(2) (w) = 1/8((1-7 z^2) log (z)-(z^2-1) (-8 log (z+1)-3+ 8 log (2)))   , 
        c s^3/6^2i  z^-s/2 H_3^(2) (w) = 1/12((7-47 z^3) log (z)+z (z^2 (49/3-27 log (3))-9 z+9) .  
           . +27 (z^3-1) log(z^2+z+1)-49/3+27 log (3))   ,  
        c s^3/6^2 i z^-s/2 H_4^(2) (w) = 1/12(-96 (z^4-1) log (2)+(19-125 z^4) log (z)+48 (z^4-1) log (z+1) .  
           . +48 (z^4-1) log(z^2+1)+1/6 (z-1) (z+1) (z (223 z-128)+223))   .

In order to compare with (<ref>), we should set z= exp ( i w) at the end of computation.


A main problem here is to regularize the divergence in the integral with  g_2[w_1 ,w_2], which can be written as

    18 i y^-1/2/c s∫_1^y dy_2 ∫_1^y_2 dy_1
    	(1 - y_1) (y_2 - y) (y_1 - y_2)/(y_2 - y_1)^4

in terms of y,y_1,y_2 in (<ref>).
Notice that this is proportional to the 1/c correction in
. ⟨ W_-1/2 (y , 1) ⟩|_𝒪(c^-1) with the AdS_3 background. Applying the regularization for the case, we find

    18 i y^-1/2/c s∫_1^y dy_2 ∫_1^y_2 dy_1
    	(1 - y_1) (y_2 - y) (y_1 - y_2)/(y_2 - y_1)^4 - 2 ϵ
           
    	 =18 i y^-1/2/c s[ 1-y/4 ϵ-1/4 (y-1) (2 log (y-1)+1)] + 𝒪 (ϵ)   .

However, this computation is too naive since we are actually working on the cylindrical coordinate w but not on  y (≡exp ( i sw)). Using the relation (see appendix <ref> for some discussions)

    . ⟨ W_-1/2 (w , 0) ⟩|_𝒪(c^-1)
    	= . [ ( ∂ w/∂ y)^-h_1,2⟨ W_-1/2 (y , 1) ⟩]  |_𝒪(c^-1)  ,

we find

    18 i y^-1/2/c s∫_1^y dy_2 ∫_1^y_2 dy_1
    	(1 - y_1) (y_2 - y) (y_1 - y_2)/(y_2 - y_1)^4 
           ∼ - 9 i y^-1/2/ c s (y-1) ( log (y-1) - 12 log (y)  +  δ )  ≡ H_s^(3) (w)  .

Here, we have  removed the divergent term by changing the overall normalization of the open Wilson line. Moreover, we introduce δ by using the ambiguity at the ϵ^0 order.
For the comparison with (<ref>), we should set y = exp (i s w) as above.


The 1/c order correction to the tree level result in (<ref>) is given by the sum of three contributions as ∑_a=1^3 H_s^(a) (w), which are given in (<ref>), (<ref>), and (<ref>).
We have checked for explicit values of s (i.e., for s=2,3,…,50)
that the sum reproduces the 1/c order expression of (<ref>) by setting the parameter as δ = - log (s).




§ CONCLUSION AND OPEN PROBLEMS



We evaluated the networks of open Wilson lines in the sl(N) Chern-Simons gauge theory with N=2,3 including quantum corrections. We compared the results with the 1/c expansions of correlators and conformal blocks in the W_N minimal model (<ref>). In order to treat divergences associated with loop diagrams, we adopted the renormalization prescription proposed in <cit.>.
In the previous paper, we examined the two and three point functions in (<ref>) with general light operators for N=2 and with specific operators for N=3 by applying the prescription. 
We first extended the analysis by evaluating the correlators in (<ref>) with general light operators for N=3. Applying the method, the identity W_3 block was computed from the product of two open Wilson lines, and the result reproduced the CFT one obtained in appendix <ref>. We further studied general conformal blocks with N=2. In particular, we obtained the all-order expressions in z expansion at the 1/c order, which are available in <cit.> except for a part. 
We also made some comments on the N=3 extensions.
For a heavy-light correlator, we examined an open Wilson line in a conical geometry, where we utilized the analysis in <cit.> for the quantum effects of the geometry. The analysis may be understood as a bulk interpretation of the CFT computations in <cit.>. 

We concentrated on the Virasoro case for the 1/c corrections of general light-light blocks and heavy-light correlators, but it is desired to analyze the W_3 case in a systematic way.

For the general W_3 blocks at the leading order in 1/c, we explained that our formulation nicely combines previous analyses in <cit.> from CFT and in <cit.> with open Wilson lines.
We thought that it is straightforward to extend the analysis beyond the leading order just like the N=2 examples, but it does not seem to be the case.
A complication may arise from the use of the X-basis as, for instance, the bra and ket states are expressed in a quite different way as in (<ref>) and (<ref>).
In appendix <ref>, we analyze the 1/c correction of a simple example using the technique of <cit.>, but we would like to develop our method with the X-basis for more systematic analysis. 

The analysis of a heavy-light correlator is also much simpler with N=2.
For general N, we may use the results in <cit.>
on the quantum effects of conical spaces.  However, it is non-trivial to regularize divergences from loops associated with the open Wilson line in a non-trivial background. 
For N=2, a four point function in the heavy-light limit can be mapped to a two point function by applying a coordinate transformation as explained in <cit.>. Thus, we can apply our regularization prescription in the new coordinate system and come back to the original setup, see appendix <ref>. However, we do not know whether we have such a nice transformation for general N. 

The current analysis can be applied to other setups.
As a next project, we would like to consider supersymmetric extensions.
These extensions are important to connect higher spin theory to superstring theory as in <cit.> 
with 𝒩=3 supersymmetry and <cit.> for 𝒩=4 supersymmetry.
As an example, the 𝒩=2 version of <cit.> was proposed in <cit.>, and the large c regime of the 𝒩=2 W_N+1 minimal model is claimed to be described by the Chern-Simons theory based on sl(N+1|N) <cit.>.
Therefore, superconformal blocks are expected to be computed by the networks of open Wilson lines in a Chern-Simons theory based on a superalgebra.


An important point of our formulation is to use open Wilson lines instead of bulk scalar propagators.
It is better to use bulk scalars if we want to do higher dimensional extensions, see, e.g.,  <cit.> for a recent argument. However, Wilson lines are gauge invariant objects and should be useful in examining the properties of geometry in terms of Chern-Simons theory. In particular, a Wilson line was used to compute the entanglement entropy of boundary theory in a holographic way <cit.>, which is a higher spin extension of <cit.>, see also <cit.>.
We expect that our formulation is useful to extract non-trivial information on the quantum effects of non-trivial geometry.






 §.§ Acknowledgements


We are grateful to Changhyun Ahn, Andrea Campoleoni, and Euihun Joung for useful discussions.
We thank the organizers of “East Asia Joint Workshop on Fields and Strings 2017, KEK Theory Workshop 2017,”  and Y. H. thanks the organizers of the workshop “New Ideas on
Higher Spin Gravity and Holography” at Kyung Hee University, Seoul for their hospitality.
The work of Y. H. is supported by JSPS KAKENHI Grant Number 16H02182.




§ CFT CALCULATIONS



In this appendix, we first provide the computation of the four point function (<ref>) with Λ = n ω_m and Ξ = f, i.e.,
 

    G(z) ≡⟨_(n ω_m;0)(∞) _(n ω_m;0)(1) _(f;0)(z) _(f;0)(0) ⟩  .


We adopt the Coulomb gas method in <cit.>, see also <cit.>.
We then compute the three point functions in (<ref>) as


    ⟨_(n ω_m;0)(z_1) _(n ω_m;0)(z_2) J^(s)(z_3) ⟩


from the conformal block decomposition of the four point function (<ref>) by following the strategy in <cit.>.




 §.§ Coulomb gas method


We start from N-1 free scalar fields ϕ≡ϕ^i   (i=1,…,N-1) with the normalization of two point functions as


    ⟨ϕ^i(z,z̅) ϕ^j(0,0) ⟩ = -2δ^ijlog|z|^2   .


We use the vertex operators


    V_ = : e^i·ϕ :   ,    = -α_+Λ_+ -α_-Λ_-


with the conformal weight h_=^2 and α_± in (<ref>).
The su(N) weight vectors Λ_± are decomposed as


    Λ_± = ∑_i=1^N-1λ_i^±ω_i


with the Dynkin labels λ_ i ^± and the fundamental weights ω_i. 
The Dynkin labels are related to the numbers n_i^± introduced in section <ref> as 
λ^±_i = n^±_i - n_i+1^±.
In particular, we set


    _(f;0) = -α_+ω_1  ,  _(nω_m ; 0) = -α_+nω_m

for our computations.

We introduce a background charge, which modifies the energy momentum tensor as


    T(z) = -1/4∂ϕ·∂ϕ + i α_0 ·∂^2 ϕ  .


Here α_0 ≡α_0 ρ with α_0 = α_+ + α_- and ρ  (=∑_i ω_i) is the Weyl vector of su(N).

Then, the conformal weight h_ is shifted as


    h_=·(-2α_0)   .


This is invariant under the exchange ↔ 2α_0-,
[The conjugation operation  replaces λ^±_i by λ^±_N-i.]
which implies that we should identify the operator V_ with V_2α_0-. 
In terms of weight vectors, the conformal weight becomes (<ref>), and in particular,


    h(f;0)  = N-1/2N( 1+N+1/k+N)  ,      h(n ω_m;0) = nm(N-m)/2N( n + N+n/k+N)

with finite k,N.


In the free theory, the correlator of the form

    ⟨ V__1(z_1) … V__n(z_n) ⟩

is non-zero when the charges are conserved.
In the presence of background charge, the conservation  rule is  shifted as


    ∑_i=1^n _i = 2 α_0  .


In order to satisfy this condition and compute a correlation function, we introduce the screening operator


    Q ≡∮ dz V_(z)   ,


where  is chosen to satisfy h_ = 1.
The condition comes from the fact that the dimension of Q should be zero in order not to break conformal properties of the correlation function. 
From this condition, we find


    = α_±e_i   ,


where e_i is one of the simple roots of su(N).

For our calculations, it is useful to use the following relations about the inner products as


    ω_i ·ω_j = (N-i)j/N  (i ≥ j)  ,  ω_i ·e_j = δ_ij ,
       e_i ·ρ = 1  ,  e_i ·e_j = A_ij ,


where A_ij is the Cartan matrix of su(N).





 §.§ Four point function


We consider the four point function in (<ref>). From the mapping (<ref>), the correlator is written as


    G(z) = ⟨ V_2α_0+α_+ nω_m(z_∞) V_-α_+ nω_m(1) V_-α_+ ω_N-1(z) V_-α_+ ω_1(0) (∏_i=1^N-1∮ dt_i V_α_+e_i(t_i)) ⟩  .


Here we have introduced z_∞, which will be set as z_∞ = ∞ at the end of the computation. 
We have changed the first operator V_→ V_2α_0- and
inserted the screening operators to satisfy (<ref>).

Using the simple form of the free field correlator as 

    ⟨ V__1(z_1) … V__n(z_n) ⟩ = ∏_i<j |z_i-z_j|^4_i·_j ,

the holomorphic part of G(z) is written as


    G(z) =(z_∞-1)^γ_1 (z_∞-z)^γ_z z_∞^γ_0 (1-z)^2α_+^2nm/N z^2α_+^21/N
         ×∮ dt_1 ⋯∮ dt_m-1 t_1^-2α_+^2[ ∏_i=1^m-1 (t_i-z_∞)^4α_0α_+ (t_i-t_i+1)^-2α_+^2] 
         ×∮ dt_m (t_m-z_∞)^4α_0α_+ + 2nα_+^2 (t_m-1)^-2nα_+^2
         ×∮ dt_m+1⋯∮ dt_N-1( t_N-1 - z)^-2α_+^2[ ∏_i=m+1^N-1 (t_i-z_∞)^4α_0α_+] [ ∏_i=m^N-2 (t_i-t_i+1)^-2α_+^2]   .


Here, we can compute γ_1,γ_x,γ_0 explicitly, but they are not necessary for our purpose. Using the relation


    ∮ dt_i ∏_j=1^3 (t_i-s_j)^2β_j∝ (s_1-s_2)^β_1+β_2-β_3(s_2-s_3)^β_2+β_3-β_1(s_3-s_1)^β_3+β_1-β_2


for ∑_j=1^3β_j = -2, we can integrate over all variables without t_m.
After the integration, we arrive at


    G(z) ∝| z^2α_+^21/N (1-z)^2α_+^2nm/N∮ dt_m t_m^a (t_m-1)^b (t_m-z)^c |^2


with


    a = m-1-2mα_+^2   ,   
    b = -2nα_+^2   ,    c= N-m-1-2(N-m)α_+^2  .


As in (9.88) of <cit.>, the final result should be of the form


    | ∮ dy y^a (y-1)^b (y-z)^c |^2  
            = sin(π b)sin(π(a+b+c))/sin(π(a+c)) |I_1(z)|^2 + sin(π a)sin(π c)/sin(π(a+c)) |I_2(z)|^2   ,


where I_i(z) are


    I_1(z)    = Γ(-a-b-c-1) Γ(b+1)/Γ(-a-c)_2F_1(-c, -a-b-c-1; -a-c; z)  ,
    
    I_2(z)    = z^1+a+cΓ(a+1) Γ(c+1)/Γ(a+c+2)_2F_1(-b, a+1; a+c+2; z)  .


Applying this result and using the relation


    _2F_1(a, b; c; z) = (1-z)^c-a-b_2F_1(c-a, c-b; c; z)  ,


we obtain (<ref>) with (<ref>).






 §.§ Identity block expansion and higher spin charges


We consider the holomorphic part of the identity block as

    z^-2 h(f;0)  (1-z)^nmk+N+1/N(k+N)_2F_1(n+n/k+N,-m/k+N;-N/k+N;z )  .


We obtain the three point functions in (<ref>) including 1/c corrections by applying the method in <cit.>. We first decompose the four point function as


    |z|^4 h(f;0)  G(z) = 𝒱_0(z) + ∑_s=3^∞ C_f^(s) C_n ω_m ^(s)𝒱_s(z) + …  .


Here 𝒱_0(z)  and 𝒱_s(z) are the identity Virasoro block and the Virasoro block of the spin s current, and their expansions in 1/c are given in (<ref>) and (<ref>) with h_p = s. The coefficients C^(s)_Λ are defined as


    C^(s) _Λ≡⟨𝒪̅_(Λ ; 0)𝒪_(Λ ; 0)  J^(s)⟩/⟨ J^(s) J^(s)⟩^1/2⟨𝒪̅_(Λ ; 0)𝒪_(Λ ; 0)⟩  ,


which is expanded in 1/c as


    C^(s)_Λ = c^-1/2[ C_Λ, 0^(s) +c^-1C_Λ,1^(s) +𝒪(c^-2) ]  .


On the other hand, from  (<ref>),

the expansion of (<ref>) in 1/c is obtained as


    |z|^4 h(f;0)  G(z)  
       ∼ 1+ (1-N^2)/c∑_l=1^∞( -nm/l + n/Γ(m)Γ(1+N)Γ(l+m)/lΓ(l+N)) z^l + 1/c^2∑_l=2^∞ f_c;nω_m^(l) z^l   ,


where we have defined


    f_c;nω_m^(l)/(1-N^2)^2
       = nN^2Γ(N)Γ(l+m)/lΓ(m)Γ(l+N)[ ∑_j=1^lN/N+j-1 - ( m ∑_j=1^l 1/j+m-1 -n H_l-1) - 2N^2+2N+1/N(N+1)] 
       - n^2m NΓ(N)/Γ(m)∑_j=1^l-11/l-jΓ(j+m)/jΓ(j+N) + nm/l( 2N + 1/N+1 +nmH_l-1)  .


Comparing the 1/c and z expansions in the both sides of (<ref>), we obtain constraints for the coefficients C^(s)_Λ.


We find the explicit form of the coefficients C^(s)_Λ by solving the constraint equations.
From the coefficients of z^l, constraints at the leading order in 1/c are given as


    ( -nm/l + n/Γ(m)Γ(1+N)Γ(l+m)/lΓ(l+N)) =  (Γ(l))^2/1-N^2∑_s=2^l Γ(2s) C_f,0^(s) C_nω_ m,0^(s)/(Γ(s))^2 Γ(s+l) (l-s)!  ,


where

    C_f,0^(2) C_nω_ m,0^(2)= nm/2 (1-N)(m-N)

is read off from (<ref>).
The first few solutions to the constraint equations are


    C_f,0^(3) C_nω_ m,0^(3)= nm(1-N)(m-N)(2m-N)/6(N+2) ,
       C_f,0^(4) C_nω_ m,0^(4) =nm (1-N) (m-N) (5 m^2-5 m N+N^2+1)/20 (N+2) (N+3) ,
       C_f,0^(5) C_nω_ m,0^(5) =nm(1-N) (m-N) (2 m-N) (7 m^2-7 m N+N^2+5)/70 (N+2) (N+3) (N+4) .


Using the definition of the coefficients C_Λ,0^(s) in (<ref>) with (<ref>) and the normalization of currents in (<ref>) with (<ref>), we rewrite the three point functions as

    . w^(s) (Λ , 0) |_𝒪(c^0)≡. ⟨𝒪̅_(Λ ; 0)𝒪_(Λ ; 0) J^(s)⟩|_𝒪(c^0) =
    Γ (N+s)/(1-N) Γ (N+2) C_f,0^(s) C_Λ,0^(s)  ,

which leads to

    . w^(3) (nω_ m , 0) |_𝒪(c^0)= 1/6 m n (m - N) (2 m - N)   , 
       . w^(4) (nω_ m , 0) |_𝒪(c^0) = 1/20 m n (m-N) (5 m^2-5 m N+N^2+1)   ,
       . w^(5) (nω_ m , 0) |_𝒪(c^0) = 1/70 m n (m-N) (2 m-N) (7 m^2-7 m N+N^2+5)   .

We can see that the results with s=3,4 and general n,m are the same as (<ref>) and (C.3) in <cit.>.

At the next order in 1/c, constraint equations for s=3,4,5 are given as

    f_c;nω_ m^(3) = f_c;nω_ m^(2) + ( C_f,0^(3)C_nω_ m,1^(3) + C_f,1^(3)C_nω_ m,0^(3))  , 
       f_c;nω_ m^(4) = f_c;nω_ m^(2)9/10 + (1-N)^2/8(1+N)^2 + 1-N/10(1+N)^2 + 1/50(1+N)^2
         + ( C_f,0^(3)C_nω_ m,1^(3) + C_f,1^(3)C_nω_ m,0^(3))3/2 + ( C_f,0^(4)C_nω_ m,1^(4) + C_f ,1^(4)C_nω_ m,0^(4))  , 
       f_c;nω_ m^(5) = f_c;nω_ m^(2)4/5 + (1-N)^2/4(1+N)^2 + 1-N/5(1+N)^2 + 1/25(1+N)^2
         + ( C_f,0^(3)C_nω_ m,1^(3) + C_f,1^(3)C_nω_ m,0^(3))12/7 + ( C_f,0^(4)C_nω_ m,1^(4) + C_f,1^(4)C_nω_ m,0^(4))·2 
        + ( C_f,0^(5)C_nω_ m,1^(5) + C_f,1^(5)C_nω_ m,0^(5))+ C_f,0^(3)C_nω_ m,0^(3)[1/21-N/1+N+6/7(1+N)+18/49(1-N^2)]  .


From the z^3 order constraint, we obtain


    C_nω_ m,1^(3)/C_nω_ m,0^(3)   = N^3 +3nN^2 -3N -6/N+2 -3n +4   ,


which reproduces the 1/c order term in (<ref>).
From the z^4 order constraint, we find


    C_nω_ m,1^(4)/C_nω_ m,0^(4)=1/40nm (N-1) (N+1)^2 (N+2) (N+3) (m-N) (5 m^2-5 m N+N^2+1)
       [4 (-25 nm^4 (N-1)^2 (N+1)^3 (n (N-8) (N+2) (N+3)-2 (N (2 N (N+8)+29)+6))) . 
       +50 nm^3 (N-1)^2 N (N+1)^3 (n (N-8) (N+2) (N+3)-2 (N (2 N (N+8)+29)+6))
       -5 nm^2 (N-1)^2 (N+1)^3 (n (N+2) (N+3) (N^2 (5 N-52)-2)-2 (N (6 N (N (2 N (N+8)+29)+7)+17)+6))
       -10 nm (N-1)^2 N (N+1)^3 (n (N+2) (N+3) (6 N^2+1)+N (N (N (2 N (N+8)+29)+12)+17)+6)
       .+(N+2)^2 (N+3)^2 (7-5 N)^2]
       +1/40[-40 N^3-290 N^2-60 N-3780/N-3+320/N-2-47/N-1+12/(N-1)^2+27/N+1+12/(N+1)^2+240/N+2+1440/N+3-1610] 
      .


In the same way, we can obtain the expression of C_nω_ m,1^(5) / C_nω_ m,0^(5), but the result is too complicated to write down here.




§ SL(3) GENERATORS



One of the aims in this paper is to generalize the analysis in <cit.> by examining correlators with general light operators for N=3.
We parametrize the representation of the operator by Dynkin labels as (λ_1,λ_2 ; 0).
For the generalization, we need the expressions of sl(3) generators in terms of X = (x,y,w) and the highest and lowest weight states denoted as ⟨lw| X ⟩ and ⟨ X | hw⟩, respectively. 
For the sl(3) generators, we use the expressions in <cit.> as

    f_0^1 = - ∂/∂ x  ,   
    f_0^2 = - ∂/∂ y - x ∂/∂ w  ,   
    f_0^3 = -∂/∂ w  , 
       h_0^1 = - λ_2 + 2 x ∂/∂ x - y ∂/∂ y + w ∂/∂ w  ,   
    h_0^2 = - λ_1 - x ∂/∂ x + 2 y ∂/∂ y + w ∂/∂ w  , 
       e_0^1 = - λ_2 x + w ∂/∂ y + x^2 ∂/∂ x + x w ∂/∂ w - x y ∂/∂ y  ,   
    e_0^2 = - λ_1 x - w ∂/∂ x + y^2 ∂/∂ y  , 
       e_0^3 = - ( λ_1 + λ_2 ) w + λ_1 x y + x w  ∂/∂ x + y w ∂/∂ y + w^2 ∂/∂ w - x y^2 ∂/∂ y  .

The general states are expressed by monomials of the form x^r y^s w^t.
The lowest weight state is represented as
⟨ X | lw⟩ = 1 .
The state is annihilated by the lowering operators f_0^a (a=1,2,3),
and the eigenvalues of the Casimir generators h_0^i (i=1,2) are ( - λ_2 , - λ_1).
This implies that we should use (see also <cit.>)

    ⟨lw | X⟩ = δ (x) δ (y) δ (w)   .

The highest weight states are annihilated by e^a_0 (a=1,2,3), and  the eigenvalues of h_0^i  are ( λ_1 , λ_2).  These conditions can be solved by

    ⟨ X | hw⟩ =
    2^λ_1 + λ_2 (x y - w)^λ_1 w^λ_2  ,

where the overall factor is chosen such that the expression in (<ref>) becomes simpler.



For the application to higher spin gravity, it is important to decompose the sl(3) generators in terms of embedded sl(2). 
Here, we choose

    V_1^2 = - f_0^1 - f_0^2   ,    V_0^2 = h_0^1 + h_0^2   ,   
    V_-1^2 =  2 ( e_0^1 + e_0^2 )   ,

and

    V_2^3 =  f_0^3   ,   
    V_1^3 = - 1/2 ( f_0^1 - f_0^2 )   ,    V_0^3 = 1/3 ( h_0^1 - h_0^2 )   ,   
    V_-1^3 =   - e_0^1 + e_0^2   ,   
    V_-2^3 = 4  e_0^3   .

We can show that the commutation relations

    [V_m^2 , V_n^2] = (m-n) V_m+n^2  ,   
    [V_m^2 , V_n^3] = (2 m - n) V_m +n^3  , 
        [V_m^3 , V_n^3] = - 1/12 (m - n) (2 m^2 + 2n^2 - m n - 8)  V_m+n^2

are satisfied.

For the manipulation of Wilson line operators, we need some algebraic computations.
In (<ref>) we may use

    e^ z V_1^2⟨ X|  lw⟩
    = 2^λ_1 + λ_2
    ( (x + z) (y + z) -  (12 z^2 + z x + w))^λ_1
    ( 12 z^2 + z x + w)^λ_2  ,

where we have utilized the following formula:

    e^ z V_1^2 f(X) = e^ z ( ∂/∂ x +  ∂/∂ y + x ∂/∂ w) f(X)  =  e^( z^2/2 + z x ) ∂/∂ w e^z  ∂/∂ ye^z  ∂/∂ x  f(X)   .

Similarly, we can compute

    e^z V_1^2 V_-1^2 e^ - z V_1^2   = - 2 [ λ_2 x_z + λ_1 y_z  + (x_z y_z - w_z  - y_z^2 ) ∂/∂ y
                + (w_z- x_z^2) ( ∂/∂ x - z  ∂/∂ w) - x_z w_z∂/∂ w]    , 
    
    e^zV_1^2 V_-2^3 e^ - z V_1^2   =  - 4 [ ( λ_1 + λ_2 ) w_z - λ_1 x_z y_z  - x_z w_z( ∂/∂ x -z ∂/∂ w)  
                + (x_z y_z^2 - y_z w_z) ∂/∂ y - w_z^2 ∂/∂ w]

with

    x_z = x + z     ,    y_z = y + z     ,    w_z = w + z x + 12 z^2    ,

which are used for (<ref>).




§ SIMPLE EXAMPLE OF GENERAL W_3 BLOCK



In the main context, we examined general blocks with the Wilson line method for N=2 up to the next leading order in 1/c and for N=3 at the leading order in 1/c.
In this appendix, we examine the 1/c correction of the general W_3 block for (<ref>) with N=3 and n=m=1. For the simple case, it is not difficult to apply the technique in <cit.> for algebraic calculations, even though our approach with the X-basis should be useful for more complicated examples with generic representations just as the leading order analysis in subsection <ref>.


Similarly to the N=2 case, we start by examining the three point function

    ⟨𝒪_(f̅;0) (z) 𝒪_(f;0) (0) 𝒪_(adj;0) (1) ⟩∝1/(1 -z)^ h_adjz^2 h_f - h_adj  ,

where adj represents the adjoint representation of su(3).
The conformal dimensions are

    h_f≡ h(f; 0) = - 1 - 32/c + 𝒪(c^-2)   ,   
    h_adj≡ h(adj; 0) = - 2 - 72/c + 𝒪(c^-2)   ,

see (<ref>).
In the Wilson line network of (<ref>) with n=3, we set (z_1,z_2,z_3)=(z,0,1) and (Λ_1,Λ_2,Λ_3) = (f̅ , f , adj  ).
Putting z_0=z_3 =1, we compute the 1/c corrections from the diagrams in figure <ref>. 
For the leading order in 1/c, we use

    q^(i)_j ≡⟨ e_j | e^z_0i V_1^2 | e_1 ⟩_i  = δ_j^1 - √(2) z_0iδ_j^2 + z_0i^2 δ^3_j   , 
       q̅_(i)^j ≡⟨e̅_j | e^z_0i V_1^2 | e̅_3 ⟩_i  = δ_j^3 + √(2) z_0iδ_j^2 + z_0i^2 δ^1_j

by adopting the conventions in <cit.>.
For examples, |e_j ⟩_i and |e̅_j ⟩_i denote the states in the fundamental and anti-fundamental representations, which are used to construct the general representation Λ_i.
As a singlet ⟨ S |, we first pick up the adjoint representation among Λ_1 ⊗Λ_2 and then construct a singlet with Λ_3 = adj.
This choice of singlet leads to 

    . ⟨ G_3 (Λ_i | z_i) ⟩|_𝒪(c^0) = q̅_(1)^i q^(2)_j q̅_(3)^j q^(3)_i  = (1-z)^2   ,

which reproduces the leading order term in (<ref>).
For the next leading order in 1/c,
we define

    q^(i;s)_j (z')  ≡6/c⟨ e_j |e^ (z_0 - z') V_1^2 V^s_-s+1 e^ (z' - z_0) V_1^2 e^z_0i V_1^2 | e_1 ⟩_i    , 
       q̅_(i;s)^j (z')≡6/c⟨e̅_j | e^ (z_0 - z') V_1^2 V^s_-s+1 e^ (z' - z_0) V_1^2 e^z_0i V_1^2 | e̅_3 ⟩_i   , 
       q^(i;s,s)_j (z',z”)  ≡(  6/c)^2 ⟨ e_j |e^ (z_0 - z') V_1^2 V^s_-s+1 e^ (z' - z_0) V_1^2e^ (z_0 - z”) V_1^2 V^s_-s+1 e^ (z” - z_0) V_1^2e^z_0i V_1^2 | e_1 ⟩_i    , 
       q̅_(i;s,s)^j (z',z”)≡(  6/c)^2 ⟨e̅_j | e^ (z_0 - z') V_1^2 V^s_-s+1 e^ (z' - z_0) V_1^2e^ (z_0 - z”) V_1^2 V^s_-s+1 e^ (z” - z_0) V_1^2 e^z_0i V_1^2 | e̅_3 ⟩_i   .

Just like the leading order, contributions at order 1/c come from the integrals as

    ∫_z^1 d z_2 ∫_z^z_2 d z_1  q̅_(1;s,s)^i (z_2,z_1)q^(2)_j  q̅_(3)^j q^(3)_i  ⟨ J^(s)(z_2) J^(s) (z_1) ⟩  , 
       ∫_1^0 d z_2 ∫_1^z_2 d z_1  q̅_(1)^i q^(2;s,s)_j  (z_1,z_2)  q̅_(3)^j q^(3)_i  ⟨ J^(s)(z_2) J^(s) (z_1) ⟩  , 
       ∫_z^1 d z_2 ∫_1^0 d z_1   q̅_(1;s)^i (z_2)q^(2;s)_j (z_1)   q̅_(3)^j q^(3)_i  ⟨ J^(s)(z_2) J^(s) (z_1) ⟩

with s=2,3 and (<ref>) with (<ref>). The sum of them is computed as

    . ⟨ G_3 (Λ_i | z_i) ⟩|_𝒪(c^-1) = 1/c (1 - z)^2 (72 log (1 - z )- 8 log (z))

for the ϵ-independent part. With the shifts of conformal weight in (<ref>), we can see that this is the result expected from the conformal symmetry as in (<ref>).


The four point block for (<ref>) with N=3 and n=m=1 can be examined in a similar manner.
Since we have already examined the identity block in subsection <ref>, we focus on the W_3 block with the exchange of adjoint operator in (<ref>).
The expression in 1/c expansion is

    (1-z)^2-z^2/3 + 1/c[ ((1-z)^2-z^2/3) (64 log (1-z)-8 log (z))
           +24 z (1-z)^2  -8 (1-z)^2 ∑ _n=2^∞ (n-1) (3 H_n-2+1/n-3) z^n] + 𝒪(c^-2)   .

In the Wilson line network (<ref>) with n=4, we set (z_1,z_2,z_3,z_4)=(z,0,∞,1) and  (Λ_1,Λ_2,Λ_3 , Λ_4) = (f̅, f,f̅, f ).  
We may fix z_0 = z_4 =1 as in figure <ref>.
For the singlet ⟨ S |, we first pick up the adjoint representations in Λ_1 ⊗Λ_2 and Λ_3 ⊗Λ_4 and then construct a singlet from the product of the two adjoint representations. 
At the leading order in 1/c, we find

    . ⟨ G_4 (Λ_i | z_i) ⟩|_𝒪(c^0) = q̅^i_(1) q^(2)_j q̅^j_(3) q^(4)_i - 1/3q̅^i_(1) q^(2)_i q̅^j_(3) q^(4)_j = (1-z)^2 - z^2/3  ,

which reproduces the leading order term in (<ref>). Here, we should notice that the products q̅^i_(1) q^(2)_j q̅^j_(3) q^(4)_i and q̅^i_(1) q^(2)_i q̅^j_(3) q^(4)_j are the same as the identity blocks for the z → 1 and z → 0 channels, respectively. Since the structure of contractions among  indices does not change even at the next leading orders in 1/c, the computation for the Wilson line network reduces to 

    ⟨ G_4 (Λ_i | z_i) ⟩  = ⟨ W_f (z;1) W_f (∞;0) ⟩ - 1/3⟨ W_f (z;0) W_f (∞;1) ⟩

at least up to order 1/c.
Using (<ref>) and (<ref>), the 1/c order terms of the quantity are obtained as

    . ⟨ G_4 (Λ_i | z_i) ⟩|_𝒪(c^-1)  =  1/c{[ 64 (1-z)^2 log (1-z) + 2 (1-z)^4   _2F_1(2,2;4;1-z) 
                -2/15 (1-z)^5   _2F_1(3,3;6;1-z) ] -1/3[ 64 z^2 log (z) + 2 z^4   _2F_1(2,2;4;z) 
                -2/15 z^5   _2F_1(3,3;6;z) ] }  .

The difference between the 1/c order terms in (<ref>) and (<ref>) is proportional to the leading order terms in (<ref>), which can be removed by changing the overall factor of the conformal block. Thus, we conclude that the Wilson line computation reproduces the CFT one in (<ref>) up to the 1/c order for the simple example.



§ CONFORMAL TRANSFORMATION OF CORRELATORS



In appendix A of <cit.>, it was analyzed how global conformal transformations act on  correlators from Wilson line networks at the leading order in 1/c. Here, we would like to extend the analysis to local conformal transformations.

We start from the gauge field 

    a(z) = V_1^2 + 6/c T(z) V_-1^2

as in (<ref>) with N=2 and T (z) = J^(2) (z). 
For the current analysis, we neglect the shift of coupling c_2 in (<ref>).
We consider the su(2) gauge transformation as

    a' (z) = L(z) a(z) L(z)^-1 + L(z) d/dz L(z)^-1

with

    L(z) = e^c_-1 (z) V_-1^2 e^2 log c_0 (z)  V_0^2 e^c_1 (z) V_1^2  .

We require that the a'(z) is of the form (<ref>), i.e., the coefficient of V_0^2 to be zero. 
We consider the following solution to this condition:

    c_1 (z) = 0   ,    c_-1 (z) = - c_0 (z) c_0' (z)   .

This solution reduces to (A.6) of <cit.> for global transformation induced by c_0 (z) = c z + d.

Under the gauge transformation, the open Wilson line becomes

    P exp{∫_z_i^z_f dz  [ 1/c_0(z)^2V_1^2  + c_0 (z)^2 ( 6/c T (z) + c_0 ”(z)/c_0 (z) ) V_-1^2 ] }  .

In order to set the coefficient of V_1^2 to be 1, we change the coordinate z to w with

    c_0 (z)  = (  dz/dw)^1/2  .

The open Wilson line is now rewritten as

    P exp( ∫_w_i^w_f d w     a(w)  )   ,   
    a(w) = V_1^2  +  6/c T (w)  V_-1^2

with

    T(w) = ( dw/dz)^-2( T(z) - c/12 S (w,z) )    .

Here we have defined the Schwarzian derivative as

    S(w,z) = d ^3 w / dz^3/(dw /dz) - 3/2( d^2 w / dz^2/dw/dz)^2   .


Let us consider how a two point function changes under the transformation. We first adopt the regularization in <cit.>, where we take the normal ordered prescription of T(z) in the same open Wilson line and replace the quantum number h_0 = - j by the exact conformal weight h_j. 
Then, we find

    ⟨lw | P exp(∫_z_i^z_f dz   a(z) ) | hw⟩ 
    = ⟨lw | L^-1 (z_f) P exp(∫_z_i^z_f dz   a ' (z) ) L (z_i) | hw⟩
    
    = ( dz_f/d w_f)^- h_j( dz_i/d w_i)^- h_j⟨lw | P exp(∫_w_i^w_f d w   a(w) ) | hw⟩  ,

see (A.10) of <cit.>. This is consistent with the local conformal transformation of two point function as

    ⟨𝒪_j (z_f) 𝒪_j (z_i) ⟩ =
     ( dz_f/d w_f)^- h_j( dz_i/d w_i)^- h_j⟨𝒪_j (w_f) 𝒪_j (w_i) ⟩  .

With our prescription, we regularized the open Wilson line for the two point function when the gauge field corresponds to the AdS_3 background and the coordinate z is the planar one. 
For other cases, we offer a way to regularize it such that (<ref>) is satisfied, and this is the prescription adopted in section <ref>.






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