Spin-5 Casimir Operator and Its Three-Point Functions with Two Scalars

By calculating the second-order pole in the operator product expansion (OPE) between the spin-3 Casimir operator and the spin-4 Casimir operator known previously, the spin-5 Casimir operator is obtained in the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (k,1). This spin-5 Casimir operator consisted of the quintic, quartic (with one derivative) and cubic (with two derivatives) WZW currents contracted with SU(N) invariant tensors. The three-point functions with two scalars for all values of 't Hooft coupling in the large N limit were obtained by analyzing the zero-mode eigenvalue equations carefully. These three-point functions were dual to those in AdS_3 higher spin gravity theory with matter. Furthermore, the exact three-point functions that hold for any finite N and k are obtained. The zero mode eigenvalue equations for the spin-5 current in CFT coincided with those of the spin-5 field in asymptotic symmetry algebra of the higher spin theory on the AdS_3. This paper also describes the structure constant appearing in the spin-4 Casimir operator from the OPE between the spin-3 Casimir operator and itself for N=4, 5 in the more general coset minimal model with two arbitrary levels (k_1, k_2).


Introduction
Simple examples of the AdS/CFT correspondence allow a detailed study of holography that would not be possible in a string theory setup. Recent progress along these lines is based on an examination of higher spin theories of gravity in AdS space, which include a large number of fields with spins s = 2, 3, · · · , N. In three bulk dimensions, the higher spin theory is quite simple. The massless sector is given semiclassically by the Chern-Simons action. Therefore, the graviton and its higher spin fields have no propagating modes. Two-dimensional conformal field theories are well-established quantum field theories and have a range of applications in different areas of physics. A high degree of analytical control over these CFTs can provide a rich source of CFTs with nontrivial bulk AdS duals. The simple theory is described using large N vector models. This will give rise to a controlled environment in which to study the puzzle of the emergence of a gravity dual.
The W N minimal model conformal field theory (CFT) is dual in the 't Hooft 1 N expansion to the higher spin theory of Vasiliev on the AdS 3 coupled with one complex scalar [1,2]. The CFTs have the following diagonal coset, G H , which is expressed as (1.1) The higher spin-s Casimir currents of degree s with s = 3, · · · , N can be obtained from polynomial combinations of the individual numerator SU(N) currents of spin-1 [3]. The diagonal denominator SU(N) currents of spin-1 commute with these higher spin-s Casimir currents. One of the levels for the spin-1 current is fixed by the positive integer k and the other is fixed by 1 in the numerator of the coset CFT (1.1). One of the specialties of the above coset model (1.1) is that the additional currents generated in the OPEs for the general level l in the second numerator SU(N) current become null fields for the level l = 1 and hence decouple [4]. See recent review papers [5,6] for the above duality.
The spin-3 (Casimir) current was reported in [4]. The four independent cubic terms were made from the two spin-1 currents in the numerator of the coset (1.1). The spin-4 (Casimir) current was generated [7] by calculating the OPE between this spin-3 Casimir current and itself, which consists of quartic terms and quadratic terms with two derivatives in the above spin-1 currents. The next step is to determine how one can construct the next higher spin-5 Casimir current. Because the complete form of the spin-3 and spin-4 Casimir currents is known, the OPE between the spin-3 current and spin-4 current can be calculated using the current algebra in the numerator SU(N) currents of spin-1. The spin-5 Casimir current can coset primaries. The singlet condition suggests that the zero mode of the spin-1 current with level k in the numerator of (1.1) acting on the state |(0; f ) >, where the element 0 means a trivial representation of the coset, vanishes while the zero mode of the diagonal spin-1 current in the denominator of (1.1) acting on the state |(f ; 0) > vanishes [13]. These are the eigenvalue equations, and the rightmost zero mode should be taken first when this singlet condition is applied to the multiple product (quintic, quartic and cubic terms) of zero modes. After that, the SU(N) generators appear. Then the next rightmost zero mode can be repeated with the singlet condition (each zero mode disappears and the generators with SU(N) group indices do appear) 4 . At the final stage after calculating this procedure, we are left with the eigenvalue, which has the form of a trace of the SU(N) generators contracted group indices with a range of SU(N) invariant tensors, times the state. The eigenvalue depends on N.
The relative 52 coefficients appearing in the spin-5 current are very complicated expressions in terms of N and k. This is because they arise from each coefficient function in the spin-3 current and in the spin-4 current. The latter consists of rather complicated fractional functions, whereas the former has a simple factorized form. Furthermore, the final expression for the coefficient functions is rather involved due to ordering in the composite fields. Although each 52 coefficient function is complicated, the eigenvalue equations for the spin-5 current have very simple factorized forms 5 .
In section 2, after reviewing the GKO coset construction for the spin-2 current, the generalization to the spin-3 and spin-4 currents is described.
In section 3, from the explicit expressions for the spin-3 and spin-4 Casimir currents, the second order pole of the OPE between them is obtained. The new spin-5 Casimir current is found by subtracting both the quasi primary field of spin-5 and the descendant field (of spin-5) of spin-3 Casimir current with correct coefficient functions. Section 4 analyzes the zero mode eigenvalue equations of spin-5 Casimir current in the large N limit. The three-point functions are obtained. The exact eigenvalue equations and the exact three-point functions are also described. Section 5 presents a summary of this work as well as a discussion on the future directions.
Appendices A-F provide detailed descriptions of sections 3 and 4.

Generalization of the GKO coset construction: review
The diagonal coset WZW model is given by the following coset G H , which is expressed as The numerator spin-1 currents are denoted by K a (z) of level k and J a (z) of level 1, whereas the denominator spin-1 current is denoted by J ′a (z) of level (k + 1). The standard OPE between J a (z) and J b (w) is where the level is characterized by the positive integer k 1 , which is fixed as 1 in the coset model (2.1). The other spin-1 current with level k living in the other SU(N) factor of the above coset model (2.1) satisfies the following OPE between K a (z) and K b (w), f abc K c (w) + · · · , (2.3) f abc J ′c (w) + · · · , k ′ ≡ k 1 + k 2 ≡ k + 1. (2.5) By construction, the level of the denominator current in (2.5) is the sum of each level of the numerator current. The stress energy tensor with arbitrary levels (k 1 , k 2 ) via a Sugawara construction is expressed as T (z) = − 1 2(k 1 + N) J a J a (z) − 1 2(k 2 + N) K a K a (z) + 1 2(k 1 + k 2 + N) J ′a J ′a (z). (2.6) The corresponding stress energy tensor in the coset model (2.1) can be read off from (2.6). The OPE of the stress energy tensor (2.6) with itself from the OPEs (2.2), (2.3) and (2.5) is given by where c in the fourth-order pole of (2.7) denotes the central charge in the coset (2.1) that depends on N and k and is given by . (2.8) The higher spin-3 current consists of cubic WZW currents contracted with the SU(N) completely symmetric traceless invariant d abc tensor. The relative coefficient functions were fixed by two conditions: 1) this spin-3 current transforms as a primary field under the above stress energy tensor (2.6) with levels (1, k); and 2) the OPE between this spin-3 current and the diagonal current (2.4) does not have any singular terms. Finally, the overall coefficient can be fixed by calculating the highest-order singular term in the spin-3 spin-3 OPE, which is equal to c 3 with (2.8). A higher spin-4 current can be constructed similarly, and consists of quartic, cubic (with one derivative) and quadratic (with two derivative) WZW currents. This spin-4 current can be obtained in two ways. One is to consider the most general spin-4 current and determine the various unknown coefficient functions using the above two requirements. The other is to calculate the OPE between the above spin-3 current and itself and read off the second-order pole structure. The normalization of the spin-4 current is related to the normalization of the spin-3 current and the structure constant appearing in the spin-4 current from the OPE between the spin-3 current and itself. In this case, the structure of the spin-4 current is determined completely except for the normalization constant because the full expression of the spin-3 current is known. 3 Spin-5 Casimir operator Subsection 3.1 reviews the relevant facts about the spin-3 and spin-4 currents first. This subsection will focus on the second-order pole in the OPE between the spin-3 current and the spin-4 current. Subsection 3.2 then constructs the next higher spin-5 current from the known spin-3 and spin-4 currents explicitly.

Review on the spin-3 and spin-4 currents
For later convenience, let us introduce spin-3 operators and spin-2 operators As described before, the d symbol appearing in (3.1) and (3.2) is a completely symmetrical traceless SU(N) invariant tensor of rank 3. The operators Q(z) and Q a (z) are the primary fields of spin-2 and spin-3, respectively, under the stress energy tensor defined in the second numerator SU(N) 1 factor in coset (2.1), corresponding to the first term of (2.6). Similarly, the operators, R(z) and R a (z), are the primary fields of spin-2 and spin-3, respectively, under

3)
J a (z) Q(w) = − 1 (z − w) 2 3(k 1 + N)Q a (w) + · · · , (3.4) f abc R c (w) + · · · , (3.5) K a (z) R(w) = − 1 (z − w) 2 3(k 2 + N)R a (w) + · · · . (3.6) Occasionally, some of the OPEs can be obtained from the known OPEs by taking the symmetries between the currents and levels. For example, the OPE (3.5) can be obtained from the OPE (3.3) using J a → K a (and Q a → R a ) and k 1 → k 2 without calculating (3.5) independently and vice versa. Similarly, the OPE (3.6) is related to the OPE (3.4) with J a ↔ K a and k 1 ↔ k 2 . Therefore, some OPEs and some operators are expressed with arbitrary levels (k 1 , k 2 ) rather than fixing them. The coset spin-3 primary field can be expressed as where the coefficient functions that depend on k 1 , k 2 or N were determined elsewhere [4] A 1 ≡ k 2 (N + k 2 )(N + 2k 2 ), Actually, there is an overall normalization factor B(N, k 1 , k 2 ) in (3.7), but B(N, k 1 , k 2 ) is not considered. Compared to the stress energy tensor (2.6), the spin-3 current is not the difference between the spin-3 current in the numerator and the spin-3 current in the denominator in the coset model (2.1). As described previously, by assuming the possible terms of spin-3, all the relative coefficient functions are fixed by the above two conditions. The normalization factor where the 21 coefficient functions are The last seven-terms of (3.9) newly appear and the first five-terms of (A.1) disappear. From (3.10), the precise relationships between the spin-4 current (A.1) and its different versions (3.9) and the subscripts of the h coefficients are kept unchanged.

The spin-5 current
The next step is to construct the spin-5 current from the spin-3 current (3.7) and spin-4 current (3.9). Although in principle the complete OPE can be calculated, this subsection focuses on the second order pole only to extract the spin-5 current. The OPE between the spin-3 and spin-4 currents is given by [2] where the (N, c)-dependent coefficients are Because the primary currents are not normalized properly, there is little difference from [2] in an overall factor The combination (T W (3) − 3 14 ∂ 2 W (3) )(w) ≡ Λ(w) in (3.12) is a quasi-primary field 6 . The explicit (N, c)-dependence of the structure constant C 4 33 in (3.11) was obtained from the literature [16,2] Note that due to the (N − 3) factor in (3.13), the coset model (2.1) for N = 3 does not produce the spin-4 current. The spin-5 current appearing in the OPE between the spin-3 and 6 The coefficient 3 14 comes from general formula 3 2(2s+1) , where s is the spin for the field Φ [9,12]. In other words, the OPE T (z) Λ(w) has no third order pole and the nonvanishing fourth order pole of this OPE is given by 1 14 (114 + 7c)W (3) . The numerical factor 1 14 can be obtained from the formula 1 n! n−1 x=0 with h i = 3, h j = 4, h k = 3 and n = 2 [9,12]. Note that the equation, C 3 34 = 3 4 C 4 33 holds for general N . Furthermore, for N = 4, expression (3.12) without the spin-5 current is already known in [14,9,15].
the spin-4 currents should contain the factor (N − 4) in its expression. The N should be greater than 4. This issue will be addressed later. See equation (4.33).
The spin-5 current can be obtained from 14) The stress energy tensor and the spin-3 current are given in (2.6) and (3.7). Furthermore, the relative coefficients are given in (3.11) together with (3.13). To ensure the right relative coefficients in the right hand side of (3.14), two arbitrary unknown constants can be introduced in front of η Λ and η W . For N = 3, they are vanishing by calculating the OPE between T (z) and W (5) (w) and requiring W (5) (w) is a primary field of spin-5. This is consistent with the fact that the structure constant C 4 33 in (3.13) vanishes at N = 3. For N = 4 and N = 5, these extra parameters are equal to 1 by solving the primary condition. The next step is to calculate the second order pole for the OPE W (3) (z) W (4) (w) explicitly. Appendix B presents all the detailed calculations. Appendix C describes the fully normal ordering procedure. For N = 3 or N = 4, all the calculations in Appendices B and C were checked explicitly using Thielemans package [17].
After the complicated calculations, the spin-5 current is summarized by the following 52 terms: (3.15) where the 52 coefficient functions a i are given in Appendix D (D.1). The spin-5 current (3.15) at N = 5 was checked to confirm that there are no singular terms (in the OPE between the stress energy tensor and the spin-5 current) with an order greater than 2. For consistency check, the OPE between the spin-5 current and the diagonal current was also calculated, which showed that are no singular terms when N = 4. The primary condition with N = 4 was checked until the third order singular terms. Furthermore, when N = 3, the above regularity and primary conditions (up to the second order pole) were checked explicitly. The spin-5 current consists of the first 24 quintic terms, the middle sixteen quartic terms with the f symbol (with one derivative), and the remaining twelve cubic terms with the two derivatives. The last twelve-terms can be observed from the second derivative of spin-3 current W (3) (z). The twelve quintic terms containing the δ symbol also can be seen from the composite operator, T W (3) (z). The vacuum character (7.18) of [3], which is equal to , contains the numerical factor, 6, in front of the q 5 term. This suggests that there are six spin-5 fields, ∂W (4) (z), ∂ 3 T (z), ∂T 2 (z), and W (5) (z). Among these fields, the only primary spin-5 field is given by W (5) (z). For the next higher spin-6 fields, there are eleven nonprimary fields, which can be obtained from the higher spin currents of spin s = 3, 4, 5, the stress energy tensor, and derivatives, and a single spin-6 primary current. This section describes the three-point functions with scalars for the spin-5 current found in the previous section. The large N 't Hooft limit [1] is defined as The product of the SU(N) generators has the following decompositions with δ, d and f symbols (for example, see reference [7]) A range of quintic products with three d symbols can be obtained from equation (4.2) and the traceless condition for the anti-hermitian basis. The zero modes of the current satisfy the commutation relations of the underlying finite dimensional Lie algebra SU(N). For the state |(f ; 0) >, T a corresponds to K a 0 and for the state |(0; f ) >, T a corresponds to J a 0 as follows

Eigenvalue equation
Now the zero mode eigenvalue equation of the spin 5 primary field W (5) (z) acting on the primary state (f ; 0) in the large N 't Hooft limit can be calculated using the relations in Appendix (D.3). The ground state transforms as a fundamental representation with respect to K a 0 (and as an antifundamental representation with respect to J a 0 ). For example, calculate the following eigenvalue equation corresponding to the first term of the spin-5 current How does one obtain the zero mode of the composite operator? Using the standard definition for the zero mode, (A.2) of [34], That is, the zero mode is the product of each zero mode but the ordering is reversed (the indices of a, b, c, d, and e go to the indices of e, d, c, b and a). The next step is to use the eigenvalue equation. Because the singlet condition for the state (f ; 0) is expressed as The expression (4.4) is equivalent to where the zero mode K a 0 commutes with the zero mode J b 0 and K a 0 is moved to the left. The singlet condition (4.5) is applied to the rightmost zero mode J b 0 (4.6) and the following result can be obtained This procedure can be repeated until all the J a 0 's are exchanged with the K a 0 's with the appropriate ordering. Therefore, the above eigenvalue equation (4.3) becomes In the first line of (4.8), 1 N is multiplied because the eigenvalue is needed (not the trace). Of course, the zero mode eigenvalue equation can be expressed using the zero mode J a 0 rather than K a 0 with the corresponding SU(N) generator rather than T a 7 . Here the following quantity (related to the first term of spin-5 current) can be used: where the identities of Appendix A of [7], (A.3) and (A.4), are used. Note δ aa = N 2 − 1. For N = 3, the above identity is checked explicitly. At the final stage, a large N limit (4.1) is taken. Similarly, the second-, third-, fourth-, sixth-, seventh-, ninth-, tenth-, eleventh-and twelfth-terms of the spin-5 current can be analyzed. One can proceed to calculate the remaining trace identities. For the fifth-term (and eighthterm) of the spin-5 current, the following calculation can be performed: The ddd product can be reduced to a single d and the ddf product can be written in terms of a single f using the identities involving the f -and d-tensors of SU(N). For the N = 3, the above identity has been checked explicitly. The large N behavior of (4.11) is different from 7 Similarly, the following eigenvalue equation corresponding to the ninth-term of the spin-5 current can be obtained In the first line of (4.9), the zero mode is obtained by changing the ordering of the current reversely. The singlet condition (4.5) is used in the second line. There is no change in sign because of the even number of J a 0 's. In the third line, 1 N is multiplied and the cyclic property of the trace is used. Finally, the previous result of (4.10) is used in the last line of (4.9). that of (4.10). The large N limit for the coefficient functions was not considered. Once the N behavior of these coefficient functions is included, (4.10) and (4.11) do to the final eigenvalue equations.
For the thirteenth-term (fourteenth-, · · ·, 24th-terms) of the spin-5 current, the quintic product with d symbol and δ symbol is obtained as follows: (4.12) Note that the dd product becomes a single δ symbol. The large N behavior of (4.12) is different from (4.10) or (4.11). For N = 3, the above identity has been checked explicitly.
For the 28th-term of the spin-5 current, 14) The df f product reduces to a single d symbol. Furthermore, the contributions from the 32nd-, 34th-, and 38th-terms vanish. For N = 3, 4, the above identities (4.13) and (4.14) have been checked explicitly.
Finally, the cubic product with the d symbol is calculated for the 41st-term of the spin-5 current (and 42nd-term, · · ·, 52nd-term), Let us move on the zero mode eigenvalue with one derivative corresponding to the 25thterm of the spin-5 current In the first line of (4.16), the zero mode is taken by reversing the ordering of the current as before. One then uses the following property, z m+2 , which leads to the zero mode, (∂J a ) 0 = −J a 0 . In the second line, the singlet condition (4.5) is used, and there is no sign change due to the even number of the J a 0 's. Furthermore, the previous result (4.13) is used 8 .
In this way, the nonzero zero mode contributions of all the terms in W (5)  acting on the state |(f ; 0) > in the large N limit The eigenvalue has a very simple factorized form. The same eigenvalue equation at finite N and k will be evaluated. 8 Consider the contribution from the 41st-term of spin-5 current with two derivative terms In the first line of (4.17), the zero mode is taken. The zero mode of the second derivative for the current can be obtained from the fact that In the second line, the singlet condition (4.5) is used. Note the extra minus sign because of the odd number of the current. As before, 1 N is multiplied. At the final stage, the result (4.15) is used. Appendix E describes the complete results for the eigenvalue equations in the spin-5 zero modes.

Eigenvalue equation of the zero mode of the spin-5 current
acting on the state |(0; f ) > in the large N limit Next the zero mode eigenvalue of the spin-5 primary field W (5) (z) acting on the primary state |(0; f ) > is described in a large N 't Hooft limit using the above (4.10)-(4.15) relations and Appendix (D.3). The ground state transforms as a fundamental representation with respect to J a 0 and the zero mode K a 0 has a vanishing eigenvalue equation (i.e. a fundamental representation with respect to the zero mode of the diagonal current) For example, calculate the following eigenvalue equation corresponding to the first-term of the spin-5 current where the previous result (4.4) is used. Equation (4.20) becomes the following expression using the property of the d symbol The next non-zero contribution appears in the thirteenth-term. The following nontrivial contribution can be obtained from the thirteenth-term where, as before, the trace property is used and the 1 N is multiplied to obtain the eigenvalue. The large N limit is taken at the final stage in (4.22). On the other hand, due to the N factor from (D.4), the final contribution becomes zero in the large N limit because N × N 3 = N 4 (the leading large N behavior has N 5 ).
The next nonzero contribution appears in the 25th-term as follows: In the first line, the previous relation (4.16) is used. The nonzero contribution in the 41st-term can be obtained The first line comes from (4.17). The final nonzero contribution comes from the next 42ndterm and the following is obtained: By combining (4.21), (4.23), (4.24) and (4.25) with the corresponding coefficients in the large N limit (D.4), the final zero mode eigenvalue equation can be obtained as follows: The same eigenvalue equation at finite N and k can be determined later.

Three-point functions in the large N limit
Consider the diagonal modular invariant, by pairing up identical representations on the left (holomorphic) and right (antiholomorphic) sectors. One of the primaries is given by (f ; 0) ⊗ (f ; 0) [18]. From the previous result (4.18), the eigenvalue of the spin-5 zero mode for In addition, the eigenvalue of the spin-5 zero mode for (0; f )⊗(0; f ), which is another primary, can be obtained. In this case, only four terms with no K a can survive because K a 0 |(0; f ) >= 0. Therefore, the result can be calculated from (4.26) as follows: Compared to (4.27) and (4.28), there is an overall sign change.
The three-point functions of spin-5 current with two scalars from (4.27) and (4.28) (or (4.29)) can then be expressed as The following result can be obtained by dividing these two relations (4.30) and (4.31): When λ = 1 2 , this becomes −9, which is precisely the same as the bulk computation for spin-5 in [18]. For a general λ, the bulk result in [19] for spin-5 current agrees with (4.32). Recall that the ratio (4.32) containing constant 1 without a minus sign is precisely the ratio for the spin-2 current, the ratio (4.32) containing constants 1 and 2 is exactly the ratio for the spin-3 current and the ratio (4.32) containing constants 1, 2, 3 without a minus sign is exactly the ratio for the spin-4 current. For the spin-5 current, the ratio of the three-point function is obtained from that of the spin-4 current by multiplying − (4+λ) (4−λ) . See footnote 10. The ratio of three-point function for the spin-s current can be expressed as (−1) s s−1 n=1 (n+λ) (n−λ) 9 . This issue for the finite N and k will be addressed later. 9 More explicitly, one can use the equations (4.51) and (4.53) of [19] to check the bulk result corresponding to (4.32). What does this imply in the bulk theory side? The three-point function in the bulk theory can be determined from the asymptotic behavior of a scalar field in AdS 3 as the radial coordinate goes to infinity. On the other hand, the CFT result of this paper (i.e. the equation (4.32)) indicates that for fixed spin s = 5, the λ-dependent part of this scalar field asymptotics has very simple factorized forms ± 4 n=1 (n ± λ). This provides the nontrivial information on the change in the AdS 3 scalar under the gauge transformation for the higher spin deformation in the bulk theory. Of course, this feature arises in section 4 of [19] via bulk theory computation. Therefore, one might think that the physical implication for the three-point function (4.32) is to indicate a hidden functional dependence of the change in the AdS 3 scalar (under the gauge transformation) on the deformation parameter λ where the λ = 1 2 in undeformed bulk theory.
Thus far, the normalization for the spin-5 current is not considered. The normalized spin 5 primary field W (5) (z) can be obtained using the unnormalized spin-5 current W (5) (z) (3.15) as follows: , where B is the overall coefficient function of the spin-3 current [4], C 5 34 is the structure constant and is given in [20,2]. Note the factor (N − 4) in this expression. This suggests that the SU(4) coset model does not produce a spin-5 current, as mentioned before. Moreover the structure constant C 4 33 was given in (3.13). The structure constant C 5 34 can be re-expressed as follows [2]: 4 33 is the structure constant for N = 4. Therefore, for N = 4, the structure constant vanishes. Accordingly, how does one obtain the relative coefficient factor in W (5) (z) in (4.33)? Originally, the spin-3 current has an overall factor B. One can multiply B at both sides in (3.12). For the spin-4 current, this study did not consider the structure constant C 4 33 in [7] in front of W (4) (w) of the second order pole in the OPE W (3) (z)W (3) (w). This analysis checked that for N = 4, the spin-4 current W (4) (z) W (5) (w) is on the right hand side. Once again, this 10 The eigenvalue equations can be calculated for the other currents. For the spin-4 current, the large N limit was given in (3.47) of [7]. By dividing C 4 33 with a large N limit, For the spin-3 current, by multiplying (4.22) or (4.20) of [13] by the constant A 1 (3.8) and constant B (4.33) with a large N limit, respectively, results in similar equations as follows: The λ-dependent part of the eigenvalue equation of spin-4 current in (4.34) can be obtained from that of the spin-3 current in (4.35) by multiplying (3+λ) (3−λ) and (3−λ) (3+λ) , respectively, up to the sign. This feature holds for the spin-3 and spin-2 currents. That is, by multiplying In this normalization (4.37), the previous eigenvalue equations (4.27) and (4.28) become From the description of footnote 10, the general λ dependent behavior for the eigenvalue equations of a higher spin-s current can be read off and In (4.39), the second equation can be obtained from the first equation by taking λ → −λ and vice versa up to an overall sign. Note that the ratio of unnormalized spin-5 and normalized one in the large N limit is  For the other primary state |(0; f ) >, the eigenvalue equation becomes Obviously, the previous results (4.30) and (4.31) can be obtained from (4.43) with the appropriate limit.
From (4.41) and (4.42) or (4.43), the ratio between the three-point functions at a finite N and k, can be expressed as Of course, the large N behavior in (4.32) can be observed from this general expression by taking the appropriate limit as before. In other words, (4.32) is obtained once the numerical constants, 1, 1, 2, 3, 4 in the numerator of (4.44) are ignored. This result comes from the W N coset CFT at a finite N and k, and corresponds to the ratio of three-point functions in the deformed AdS 3 bulk theory 11 . In [2,5], the minimal representation of W ∞ [µ], where the fewest number of low lying states exists, was studied. Indirectly, the eigenvalue of the zero mode of the spin-5 current, x, can be expressed in terms of the spin (or conformal dimension), h, the eigenvalue of spin-3 current, w, the eigenvalue of spin-4 current, u (note that four times their spin-4 field is equal to the spin-4 current in the present study and see (A.22) of [2]), the central charge c, the constant,  [2] (where the version 2 of arXiv corrected some typographical errors that appeared in the version 1 of arXiv) contain the eigenvalue of the zero mode of the spin-5 current, x. By substituting the following quantities 11 For the lower spin currents, the eigenvalues and three-point functions can also be analyzed. For the spin-4 current (A.1) or (3.9), the following result is described in Appendix F (F.25) where d(N, k) ≡ 17k + 17k 2 + 22N + 39kN + 5k 2 N + 22N 2 + 10kN 2 is introduced. Note the presence of the factor (N − 3). All the terms in (A.1) or (3.9) contribute to the first eigenvalue, whereas three terms in (A.1) (or four terms in (3.9)) can survive in the second eigenvalue in (4.45). For the spin-3 current (3.7), the two eigenvalue equations from Appendix F (F.26) are The first term of (3.7) can contribute to the second eigenvalue of (4.46). Note the factor (N − 2) in (4.46). Finally, the spin-2 current (2.6) has the following eigenvalue equations with Appendix F (F.27) the expression (4.49) leads to the other eigenvalue in (4.42). Checking that the eigenvalues for the spin-2, 3, 4 fields written in terms of the conformal dimension, h, and the central charge c in [2] coincide with those expressed as N and k in (4.48) and (4.50) is quite simple.
The following ratios between the three-point functions for spin s = 2, 3, 4 can be obtained from (4.47), (4.46), and (4.45) At s = 5, equation (4.44) can be derived. As a spin s increases, the extra factors in the numerator and denominator appear up to the sign. This can be generalized to the arbitrary spin-s current, W (s) , whose ratio is expressed as Of course, this generalizes the fixed λ = 1 2 case in bulk theory [18] and the arbitrary λ in the boundary/bulk theory [19] to the finite N and k. In other words, for fixed spin s, each factor in the numerator of (4.52) has the extra numerical constants, 1, 1, 2, 3, 4, · · · , (s − 1). The previous results in [19], where they assumed W ∞ [λ] symmetry in the CFT computations, can be obtained if these constants are ignored.

More general coset model
The more general coset model, G H , can be described by two arbitrary levels (k 1 , k 2 ) Many additional extra currents can be expected in the general coset model (4.53). The corresponding algebra should be larger than the conventional W N algebra. For example, the explicit form of the structure constant, C 4 33 , is not known so far in the general coset model. Recall that the spin-3 and spin-4 currents with arbitrary levels (k 1 , k 2 ) are known in the general coset model. If the central charge term of the OPE between the spin-4 current with itself (which is equal to c 4 with the central charge in the general coset model) is calculated, the explicit form for the structure constant, C 4 33 , in the general coset model can be determined completely.
The explicit forms of C 4 33 for some fixed N from the Thielemans package [17] are presented. The left hand side of (3.12) is known for an arbitrary (k 1 , k 2 ) with a fixed N = 4, 5, and two arbitrary parameters in front of η Λ and η W are introduced. The spin-5 current can be written as in (3.14) with two parameters. Now the OPE between T (z) and the spin-5 current can be calculated by requiring the primary condition. The above two parameters can be fixed completely and the structure constants for a fixed N = 4, 5 with arbitrary levels are as follows: (4.54) These structure constants are reduced to the structure constants of the minimal model when k 1 is fixed to 1 (Of course, k 2 = k). Furthermore, the highest singular term of the spin-4 current (when N = 4) was checked to ensure that it behaves correctly with (4.54).

Conclusions and outlook
The spin-5 Casimir operator (3.  2). The generalization of [21] to the multiple product of d symbols is needed.
• Because the spin-4 current is known for any arbitrary level (k 1 , k 2 ), it would be interesting to calculate the eighth-order pole of the OPE between the spin-4 current and itself, as mentioned before. Once this is done, the structure constant C 4 33 in the general coset model can then be obtained. Of course, this structure constant will reduce to the previous result (3.13) in the coset minimal model (1.1) when k 1 = 1 and k 2 = k. Furthermore, for N = 4 or N = 5, one should obtain the relations (4.54).
• What of the spin-6 current? From the vacuum character computation described before, there are eleven nonprimary spin-6 currents that can be constructed from the spin-2, 3, 4, 5 currents (with some derivatives) and a single primary spin-6 current. This current might be constructed by calculating the OPE between the spin-3 current (3.7) and spin-5 current (3.15). Definitely, the independent terms will be greater than 52 terms for the spin-5 current. At least, relation (4.52) for s = 6 can be observed. Furthermore, because of this relation, one also expects that the eigenvalue equations for two coset primary states has factorized forms, as in the other lower higher spin cases. In a bulk theory point of view, the asymptotic quantum symmetry algebra W ∞ [µ] in [2] should be extended to the algebra containing the commutator between the spin-3 Laurent mode and the spin-5 Laurent mode. The previous work given in [22] (classical algebra) is expected to be helpful in obtaining this commutator.
When N = 6, the above construction should be related to the work reported elsewhere [23], where W 6 algebra was studied.
• Because the spin-3, 4, 5 Casimir currents are known for the coset model (1.1), one can also check whether the other structure constants, C 4 44 , C 5 45 in [20], can be expressed in terms of C 4 33 (3.13). The former can be obtained from the highest singular term in the OPE of the spin-4 current and itself and the latter can be obtained from the OPE of the spin-4 current and the spin-5 current as well as the OPE between the spin-5 current and itself. In principle, the other OPEs, spin-3 spin-5, spin-4 spin-4, spin-4 spin-5, and spin-5 spin-5, can be obtained and the algebra should be related to the W 5 algebra [23], where N is fixed to N = 5. Extracting several quasi-primary fields in these OPEs is nontrivial. For example, the OPE between the spin-5 current and itself provides the second order pole, where the possible quasi-primary fields of spin-8 arise. Even in the well-known quantum Miura transformations [3], the formula appears quite complicated. As mentioned in the introduction, the general procedure to obtain the correct quasi-primaries is based on previous studies [8,9,10,11,12].
• The N = 1 supersymmetric coset model can be obtained by taking one of the levels to be N in the context of that reported elsewhere [12,24,25]. • The duality in [1,2] has been generalized to the duality with the other group, which is analogous to the O(N) vector model in one dimension higher [26,27]. In [28,29], the explicit Casimir currents are constructed. In contrast to the present work, the Casimir construction with orthogonal group has the symmetric SO(N) invariant tensor of rank 2. In other words, the general N-dependence in the OPEs can be read off from the fixed several N case results.
No complicated contractions were observed between the d symbols in the SU(N) coset model.
• The large N = 4 holography was found recently in [30]. The 2-dimensional CFT has many supersymmetries. Sometimes it is useful for describing the explicit construction of Casimir operators in N = 2 superspace. For example, in the context of the N = 2 Kazama-Suzuki model. Some relevant descriptions can be found in [31,32]. Thus far, it is unknown how to construct the N = 4 extension of W 3 algebra. Among the sixteen currents (in the spin contents, they are given by four spin-1 2 currents, seven spin-1 currents, four spin-3 2 currents and a spin-2 current) in this large N = 4 superconformal algebra, only eleven currents, after projecting out four spin-1 2 currents and a spin-1 current, play an important role in higher spin theory. The possible lowest higher spin current contains the following spin contents (1, 3 2 , 3 2 , 2), ( 3 2 , 2, 2, 5 2 ), ( 3 2 , 2, 2, 5 2 ) and (2, 5 2 , 5 2 , 3) in N = 2 superspace notation. In a future study, it would be very interesting to construct the above eleven currents living in large N = 4 nonlinear algebra and the above sixteen currents that act as extra higher spin currents in the specific coset model suggested in [30]. Therefore, it would be interesting to find (11 + 16) currents explicitly in terms of the coset WZW primaries.

Appendix A Coefficient functions appearing in the spin-4 current
The spin-4 current can be obtained by calculating the second-order pole in the OPE between the spin-3 current and itself. In this calculation, the quasi-primary field of spin-4 and the descendant term of stress energy tensor of spin-4 with the correct coefficients should be subtracted from the above second-order pole. The coset spin-4 primary field is given by [7] where the coefficient functions c i are given in [7] or in Appendix A (A.5)-(A.23). As the OPE between the spin-3 current (3.7) and spin-4 current (A.1) is calculated, the identity between the d symbols should be used. Therefore, it is better to rewrite the above spin-4 current using the d symbol of rank 3. The various identities of these d symbols of rank 3 can be used.
Using the identity [7] the first term of (A.1) can be decomposed into the right hand side of (A.2), where the following identity is used [33] 3) The first term of (A.1) can be absorbed into the sixth-and eleventh-terms of (A.1) besides the derivative terms.
One can calculate similar identities as follows [7]: Checking whether the second term of (A.1) can be absorbed into the seventh-and twelfthterms of (A.1) is relatively simple. Furthermore, the f symbol term (due to the normal ordering of the currents J a (z) and K a (z)) in the first equation of (A.4) newly arises. The third term of (A.1) can contribute to the eighth-and thirteenth-terms and the two new independent terms appear. Similarly, the fourth-term of (A.1) can be absorbed into the ninth-and fourteenth-terms. In this case, the f symbol term also arises. The symmetry between the spin-1 currents is used at the final equation of (A.4). That is, the left hand side of that equation can be obtained from (A.2) by changing J a (z) to K a (z) and vice versa.
The following presents the various coefficient functions in (A.1) that appeared in [7]: (A.8) (A.10) (A.14) (A.17) Note that (c i + b i ) in [7] is replaced with c i here. Of course, in the coset model (1.1), the levels are fixed to k 1 = 1 and k 2 = k.
Appendix B Intermediate spin-5 field contents in the secondorder pole of the OPE W (3) (z)W (4) (w) The second-order pole in W (3) (z)W (4) (w) (3.14) can be calculated explicitly using the spin-3 current in (3.7) and the spin-4 current in (3.9). Consider the first term Q(z) ≡ d abc J a J b J c (z) of (3.7) and make an OPE between Q(z) and 21 terms in the spin-4 current. For the spin-4 current, only the fields that did not have a d symbol were considered. Twelve quartic terms can be obtained from the following five terms with the free indices.
After calculating the OPE between Q(z) with the five fields in (B.1), the nine OPEs between Q(z) with the remaining nine terms in the spin-4 currents are given. The following fourteen OPEs are then obtained: For the quartic terms in the left hand side, the extra d or δ symbols are multiplied to obtain the final result for the second order pole. Further simplifications for the right hand side of (B.2) is needed to analyze the zero mode eigenvalue equations. In other words, field Q a (z) should be located at the right hand side of field J a (z) by moving Q a (z) to the right. Field K a (z) can be placed to the right. This issue will be addressed in Appendix C.
Let us introduce spin-2 operator for convenience.
Consider the second term d abc J a J b K c (z) ≡ K e Q e (z) of (3.7) and calculate the fourteen OPEs as follows: . (B.4) Further simplifications for the right hand side of (B.4) with the appropriate normal ordering procedure are needed.
The second order pole for the third term d abc J a K b K c (z) ≡ J e R e (z) of (3.7) and the spin-4 current without using an explicit calculation can be obtained from the previous result (B.4) Therefore, the second order pole structures are exhausted. Further simplifications are given in Appendix C.

Appendix C Details of the fully normal ordered spin-5 operator
Thus far, the results from the previous Appendices are not fully ordered in the sense of [34].
Further simplifications are needed with the help of the rearrangement lemma.
For example, take Q(z)d abe d cde J a J b K c K d (w)| 1 (z−w) 2 from the third equation of (B.2) in Appendix B by multiplying two d symbols As stated before, the field Q a (z) in (C.2) should be moved to the right. The right hand side of (C.2) can be expressed as and using the third equation of (C.1) this (C.3) can be expressed as Owing to the antisymmetric property of the f symbol and the symmetric property of the d symbol, the first term of (C.4) vanishes and the remaining terms are The first term in (C.5) can be simplified using the identity between two d symbols The last term of (C.6) can be simplified further as follows: where the definition of Q b (z) is used. The last two terms of (C.7) can be expressed as after substituting the commutators in (C.1). Using the identity of (A.7) of [7] (the triple product ddf is reduced to a single f ), this expression (C.8) can be simplified as The field ∂J a in the first term of (C.9) should be moved to the right to yield the following: Again, the commutator can be simplified, and after a small calculation, (C.10) leads to 11) The f f term in (C.11) can be simplified further and the final fully normal ordered product becomes where the right hand side of (C.12) appears in the fourth, 47th, and 35th terms in (3.15). The final form for Q(z) d abe d cde J a J b K c K d (w)| 1 (z−w) 2 can be obtained from (C.6) and (C.12) as follows: The following commutators can be obtained from (C.18) All the commutators in (C. 15), (C.17) and (C. 19) are used to simplify the normal ordered product in Appendix B.
Although this paper did not present all the fully normal ordered products leading to the 52 terms for the spin-5 current (3.15) (due to the space limit of this paper), the complicated calculations from Appendices B and C to the final 52 terms can be performed.
In this case, each coefficient function has a very simple form after taking the large N limit.
The large N behavior of these coefficient functions can be classified as follows:  Appendix E Eigenvalue equations for the spin-5 current on the perturbative state Appendix E presents the zero mode eigenvalue equations on the 52 terms of spin-5 current with perturbative state |(f ; 0) >. At each expression, the exact N-dependent expressions appearing in (4.10)-(4.15) are assumed. The only large N limit is presented.
Appendix F Eigenvalue equations for other higher spin currents of spins s = 2, 3, 4 The eigenvalue equations for the zero mode of the higher spin-4 current can be analyzed as follows:   In particular, the following identity with (A.3) can be used to simplify (F.1)-(F.5): where the quartic product of the generators with (4.2) has the following identities The following eigenvalue equations for other state can be obtained: where d(N, k) ≡ 17k + 17k 2 + 22N + 39kN + 5k 2 N + 22N 2 + 10kN 2 . The spin-3 current case can be analyzed further as follows: where the property (4.15) is used and the coefficients in (3.8) are substituted. Finally, the eigenvalue equations for the spin-2 current can be summarized as where the property δ ab Tr(T a T b ) = −(N 2 − 1) is used.