Higher Spin Currents in the Holographic N=1 Coset Minimal Model

In the N=1 supersymmetric coset minimal model based on (B_N^{(1)} \oplus D_N^{(1)}, D_N^{(1)}) at level (k,1) studied recently, the standard N=1 super stress tensor of spins (3/2,2) is reviewed. By considering the stress tensor in the coset (B_N^{(1)}, D_N^{(1)}) at level k, the higher spin-2' Casimir current was obtained previously. By acting the above spin-3/2 current on the higher spin-2' Casimir current, its superpartner, the higher spin-5/2 current, can be generated and combined as the first higher spin supercurrent with spins (2', 5/2). By calculating the operator product expansions (OPE) between the higher spin supercurrent and itself, the next higher spin supercurrent can be generated with spins (7/2,4). Moreover, the other higher spin supercurrent with spins (4',9/2) can be generated by calculating the OPE between the first higher spin supercurrent with spins (2', 5/2) and the second higher spin supercurrent with spins (7/2,4). Finally, the higher spin supercurrent, (11/2,6), can be extracted from the right hand side of OPE between the higher spin supercurrents, (2', 5/2) and (4', 9/2).

(1) N ) at level (k, 1) studied recently, the standard N = 1 super stress tensor of spins ( 3 2 , 2) is reviewed. By considering the stress tensor in the coset (B (1) N , D (1) N ) at level k, the higher spin-2 ′ Casimir current was obtained previously. By acting the above spin- 3 2 current on the higher spin-2 ′ Casimir current, its superpartner, the higher spin- 5 2 current, can be generated and combined as the first higher spin supercurrent with spins (2 ′ , 5 2 ). By calculating the operator product expansions (OPE) between the higher spin supercurrent and itself, the next higher spin supercurrent can be generated with spins ( 7 2 , 4). Moreover, the other higher spin supercurrent with spins (4 ′ , 9 2 ) can be generated by calculating the OPE between the first higher spin supercurrent with spins (2 ′ , 5 2 ) and the second higher spin supercurrent with spins ( 7 2 , 4). Finally, the higher spin supercurrent, ( 11 2 , 6), can be extracted from the right hand side of OPE between the higher spin supercurrents, (2 ′ , 5 2 ) and (4 ′ , 9 2 ).
In this paper, we would like to construct the higher spin currents for the coset model (1.1). For example, in order to understand this duality, the three-point functions can be compared to each other. Once the higher spin currents with bosonic spins are known completely, then in principle, the three point functions can be obtained 1 . The direct construction using the Jacobi identities in [7] doesn't tell us what the central charge is. Only after the isomorphism between the Drinfeld-Sokolov reduction and the coset construction is used, the central charge in the direct construction [7] can be identified with the one in the above coset model (1.1) where the central charge c is equal to c = 3N k (2N +k −1) . Furthermore, if one considers more 1 We will determine the complete expressions for the bosonic higher spin currents of spins s = 2 ′ , 4, 4 ′ and 6. Then one can proceed the previous analysis done in different coset model [8]. The zero mode for each bosonic current acts on the vector representation. The zero mode of the spin-1 current with level k in the numerator of the coset (1.1) acting on the state |(0; v) > vanishes while the zero mode of the diagonal spin-1 current in the denominator of the coset acting on the state |(v; 0) > vanishes. For the former, the zero mode for higher spin current consists of multiple product of quadratic in the fermions of the numerator current (i.e. K a K b (z) in section 2) and for the latter, the zero mode can be written in terms of the numerator current with level k (i.e. J AB (z) in section 2) (or the combination of fermions in the numerator current (i.e. K a K b (z)) and the numerator current with level k where one index is fixed by (2N + 1) (i.e. J a2N +1 (z) in section 2)). Some identities in the trace of the generators of SO(2N ) or SO(2N + 1) can be used in the eigenvalue equations. One should take into account the fact that the first factor of the numerator group of the coset has different from other factor groups in the coset and there exist spin- 1 2 current in the second factor of the numerator group, compared to the previous results in [8]. It would be interesting to obtain the three point functions from the findings of this paper explicitly. general coset where the second level 1 is replaced by an arbitrary integer l, then the extra current will appear in general. The above isomorphism cannot be used in this case also. This is one of the reasons why we are interested in the construction of higher spin currents for the coset model explicitly.
The main procedure in this coset construction is based on the fact that once the lower higher spin currents are found, then the next undetermined higher spin currents can be generated, in principle, by calculating the operator product expansions (OPE) between the known higher spin currents. The lowest component spin-2 ′ in the N = 1 multiplet (2 ′ , 5 2 ) of (1.2) with known realization of N = 1 stress tensor is the fundamental higher spin current because this current generates all the higher spin currents. It turns out that once the spin-2 ′ current in the multiplet (2 ′ , 5 2 ) is found, then the spin-3 2 current in the N = 1 super stress tensor determines the spin- 5 2 in the above multiplet. Then one considers the OPE between the above spin-2 ′ current and the spin- 5 2 current. The first-order pole of this OPE determines the spin- 7 2 current in the N = 1 multiplet ( 7 2 , 4) of (1.2). Now one can continue to calculate the OPE between the spin-2 ′ current and the above spin- 7 2 current and it turns out that the spin- 9 2 current that is the second component in (4 ′ , 9 2 ) appears in the first-order pole of this OPE. Furthermore, the OPE between the spin-2 ′ and the spin- 9 2 current determines the spin- 11 2 current that is the first component in ( 11 2 , 6). In this way, all the half-integer spin currents can be obtained. What about the bosonic higher spin currents? They can be determined by calculating the OPE between the spin- 3 2 current of N = 1 stress tensor and any known higher spin current of half-integer spin due to the N = 1 supersymmetry.
Then how one can extract the correct primary or quasi-primary fields in the given singular terms in the OPE? It is known that the OPE of two quasi-primary fields, of spins h i and h j respectively, takes the form [9,10,11,12,13] γ ij corresponds to a metric on the space of quasi-primary fields. The structure constant C ijk appears in the three-point function between the quasi-primary fields, Φ i (z), Φ j (z) and Φ k (z).
The index k specifies all the quasi-primary fields occurring in the right hand side of (1.3). The descendant fields for the quasi-primary field Φ k (w) of spin h k are multiple derivatives of Φ k (w). The relative coefficient functions 1 n! n−1 x=0 in the descendant fields depend on the spins and number of derivatives. Since the higher spin currents can be written in terms of WZW currents, the above structure constant C ijk can be determined. For the fixed C ijk , the relative coefficient functions in the descendant fields using (1.3) can be obtained. In general, the singular terms of the OPE are written in terms of WZW currents in complicated way. It is quite nontrivial to rearrange those expressions in terms of determined (and known) higher spin currents. This rearrangement can be done using the primary or quasi-primary condition under the spin-2 current of N = 1 super stress tensor and superprimary condition under the spin- 3 2 current of N = 1 stress tensor. The details can be seen in the next section. In section 2, the fundamental OPEs between the WZW currents living in the coset model (1.1) are given and the spin-2 stress tensor and its superpartner with Sugawara construction are reviewed.
In section 3, from the observation of [5], the lowest higher spin N = 1 multiplet (2, 5 2 ) in (1.2) is obtained. The additional three higher spin supercurrents in the list (1.2) are constructed very explicitly.
In section 4, we summarize what we have found in this paper and discuss the future directions.
In Appendices, some results from the detailed calculations in section 3 are provided. The mathematica package by Thielemans [14] is used.

The GKO coset construction: Review
For the diagonal coset model [5] the antisymmetric spin-1 fields, J AB (z) with level k, and the spin-1 2 fields, K a (z) with level 1, generate the affine Lie algebra G = SO(2N + 1) k ⊕ SO(2N) 1 . The index a runs from 1 to 2N and the index A runs from 1 to (2N + 1). The number of independent fields of J AB and K a (z) are given by N(2N + 1) and 2N respectively. Note that this coset model (2.1) is a little different from the ones studied in [3,8,15,16]. The fundamental OPE of the fermion fields K a (z) is given by (2. 2) The standard OPE of the spin-1 currents is expressed as δ BC J AD (w) + δ AD J BC (w) − δ AC J BD (w) − δ BD J AC (w) + · · · . (2. 3) The diagonal spin-1 field J ′ab (z) with level (k + 1) generates the affine Lie algebra H = SO(2N) k+1 . It is expressed as J ′ab (z) = J ab (z) + (K a K b )(z). (2.4) Note that we are using the double index notation for the spin-1 current rather than a single index used in [5]. Among (2N +1)×(2N +1) matrix, the first (2N)×(2N) matrix corresponds to the spin-1 current J ab (z) and the remaining matrix elements correspond to other spin-1 current J a 2N +1 (z). The spin-2 stress energy tensor with (2.4), via the Sugawara construction in the coset model (2.1), is written as (J ′ab J ′ab )(z). (2.5) Since J AB (z) can be decomposed into J a 2N +1 (z), and J ab (z) which belongs to SO(2N) subgroup, (J AB J AB )(z) can be written as (J AB J AB )(z) = (J ab J ab )(z) + 2(J a 2N +1 J a 2N +1 )(z). (2.6) Therefore, by plugging (2.6) into the stress tensor (2.5), the spin-2 stress tensor can be expressed concisely as Note that there is no J ab J ab (z) in (2.7). One can easily check that there is no singular term in the OPE between T (z) and J ′ab (w). The OPE between the stress energy tensor T (z) and itself, from (2.2), (2.3) and (2.7), is given by ∂T (w) + · · · . (2.8) The central charge in the highest singular term of (2.8) is given by The superpartner of the spin-2 current T (z) is the spin-3 2 current G(z) [17]. They can be combined into a single N = 1 multiplet as ( 3 2 , 2) in (1.2). The OPE between G(z) and itself reads as 2T (w) + · · · . (2.10) One can construct G(z) from the WZW currents K a (z) of spin-1 2 and J AB (z) of spin-1. Since K a (z) has one index, (K a J a 2N +1 )(z) is the only candidate for G(z). The normalization for G(z) can be fixed from (2.10) and the explicit form of G(z) is given by . (2.11) One checks that the OPE between G(z) and the diagonal spin-1 current J ′ab (w) does not contain any singular terms. As we expect, it satisfies the following OPE: (2.12) The standard N = 1 superconformal algebra consists of (2.8), (2.10) and (2.12). In the next section, the fundamental OPEs (2.2) and (2.3) are used heavily and the coset stress tensor (2.7) and its superpartner (2.11) with central charge (2.9) will be used all the times. 3 The construction of higher spin supercurrents In this section, the higher spin currents will be constructed for general N explicitly from the fermion fields K a (z) of spin-1 2 and the antisymmetric spin-1 currents J AB (z).

3.1
The OPEs between the higher spin currents of spins-(2 ′ , 5 2 ) and itself Now, let us consider the stress energy tensor of the coset SO(2N +1) k SO(2N ) k [5], which is T (z) = − 1 4(k + 2N − 1) (J AB J AB )(z) + 1 4(k + 2N − 2) (J ab J ab )(z). (3.1) Note that the denominator current is simply given by J ab (z). This stress tensor T (z) obeys the following OPEs from (3.1) and (2.7): where c is the central charge of coset SO(2N +1) k SO(2N ) k and can be expressed as c = kN(−3 + 2k + 2N) (−2 + k + 2N)(−1 + k + 2N) . (3.4) The stress tensor T (z) is a quasi-primary field under the stress tensor (2.7). From (2.8) and (3.3), one can easily figure out that the following combination of T (z) and T (z) with (2.9) and (3.4) gives a spin-2 ′ primary field O 2 ′ (z) under the stress tensor T (z): . (3.5) This is because the OPE between T (z) and O 2 ′ (w) does not contain the fourth-order pole. In terms of J AB (z) and K a (z), the spin-2 current (3.5) is expressed as 2 O 2 (z) = kN 4(−2 + k + 2N)(−1 + k + 2N) 2 3 J ab J ab + 2k(−3 + 2k + 2N)K a ∂K a +2(3 − 2k − 2N)K a K b J ab − 2(−3 + k + 4N)J a 2N +1 J a 2N +1 (z). (3.6) Compared to (2.7), the first term of (3.6) appears newly. The OPE between G(z) and O 2 (w) generates a primary spin-5 2 current O 5 2 (z) under the stress tensor T (z) as follows: The explicit form of spin- −(−3 + 2k + 2N)K a ∂J a 2N +1 + 3K a (J ab J b 2N +1 + J b 2N +1 J ab ) (z). (3.8) The OPE between G(z) and O 5 2 (w) with (2.11) and (3.8) is described as (3.10) The relative coefficient 1 2 (= 2−2+2 2×2 ) in the descendant fields can be understood from the general formula in (1.3). There are no new primary fields in the right hand side of (3.10). The structure constants in (3.10) are The fusion rule can be summarized by (w) Now let us move to the OPE between O 2 (z) and O 5 2 (w) 3 . We follow the method used in [18] to find the complete structure of the OPE. The OPE between O 2 (z) and O 5 2 (w) for where the structure constants for fixed N = 2 are given by . (3.12) Moreover, the summation indices (appearing in O 2 (z), O 5 2 (z) and G(w)) a, b = 1, · · · , 4 because N = 2. For the primary field G(w) with the structure constant c g in the second-and first-order pole can be read off from (1.3). Refer to [18] for details on how to compute the relative coefficients of descendant fields. For the second-order pole in (3.11), the first term descending from G(w) is fixed. To check if there are other fields besides ∂G(w) in the second-order pole, we compute [ 4 . It turns out that this doesn't vanish implying that there should be extra fields besides ∂G(w). The spin of fields in the second-order pole should be 5 2 and the only candidate for this extra field is the primary field O 5 . Therefore, the extra field is proportional to For the first-order pole, the first-and second-term descending from G(w) and O 5 2 (w) respectively are completely fixed. The relative coefficient 2 5 in front of ∂O 5 2 (w) in the firstorder pole is also determined from (1.3). As we did for the second-order pole, to find if there are other fields besides ∂ 2 G(w) and ∂O 5 where the first line contains two terms in (3.11). It doesn't vanish and there should be extra fields. To find the extra fields in the first-order pole, let us consider the OPE between T (z) . This means that a quasi-primary field, T G(w) plus derivative terms, should be considered to cancel the fourth-order term of this OPE for the primary condition. Moreover, the OPE between G(z) and [ . This means that we should consider another quasi-primary field, GO 2 (w) plus derivative terms, to remove the third-order terms of this OPE. With the help of superprimary condition (primary under the stress tensor and the equation (3.14)), the consistent coefficients, c go (z) can be expressed as As expected, O 7 2 (z) obeys the following OPE: where O 4 (z) is a spin-4 primary field in (1.2) which will appear in (3.20) when the OPE between O 5 2 (z) and itself is computed later. To determine the structure constants in (3.11) for general N, we compute the OPE between O 2 (z) and O 5 2 (z) by hand explicitly with the help of (3.11). Explicit calculations for the thirdand second-order pole show that the coefficients c g For the first-order pole, we find (3.30) where N ′ ≡ 2N + 1 and the dummy indices run as a, b, c, d = 1, · · · , 2N. Of course, for N = 2, the expression ( } −1 − ( the right hand side of (3.30))](w) = 0 for the coefficients. The left hand side of (3.30) is given by too many WZW currents and we want to write it using the tensorial structure to simplify. We repeat this procedure for N = 3, 4, 5 6 . From the above results (3.31) and (3.32), one can easily figure out the general forms of c 1 and c 2 . For general N, the coefficients c 1 and c 2 are given by In this way one can find all coefficients in (3.30) for general N. We also put the equations of the coefficients for N = 2 in (A.  6 For the coefficient c 1 appearing in the first term of (3.30), each coefficient function can be obtained Similarly, for the coefficient c 2 appearing in the second term of (3.30), the coefficient functions for different N -values are given by Now we move to the OPE between the O 2 (z) and O 4 (z). From the computation of the OPE for N = 2, we find that this OPE takes the form At the moment, the right hand side holds for N = 2 case only. We would like to obtain this OPE for general N. In particular, the structure constants for general N.
It turns out that there is no solution for (3.40). This implies that there should be a new primary spin-4 field and that this new spin-4 field would be the superpartner of O 9 2 (z). To check this, we consider the following equation for c o ′ 24 and c ′ : As expected, the spin-4 current, O 4 ′ (z), obeys the following OPEs: where the spin-9 2 current is given by (3.35 Here the descendant fields in the 1st-order pole contain the first four terms in (3.39). The above OPEs (3.44) have higher-order poles with order n > 2 and these higher-order poles are removed by the following two quasi-primary fields: As done before, the relative coefficients of descendant fields are fixed by (1.3), and the quasiprimary, and primary fields are found by the same method we used before. No new primary field is found in this case. The field contents up to the second-order pole are the same as the one in (3.33). To find the general forms of structure constants, we continue to compute the OPE between O 5 2 (z) and O 4 (w) using the package for N = 3, 4, 5 cases. After computing the OPEs for N = 3, 4, 5, we solve the following equations for structure constants when N = 3, 4, 5: Here the 2nd-order pole and 1st-order pole are given in the right hand side of (3.49) respectively. Then we put all the results together and find the structure constants for general N.
The general expressions of structure constants in (3.49) are given by  · · ·. We have checked this up to N = 5. Therefore all the previous OPEs can be rewritten in terms of these rescaled currents.
3.3 The OPEs between the higher spin currents of spins-(2 ′ , 5 2 ) and the higher spin currents of spins-(4 ′ , 9 2 ) Now let us consider the OPE between O 2 (z) and O 4 ′ (z). The final result for N = 2 is presented first, which explains how this result can be obtained explicitly The structure of this OPE appears similar to (3.39) except that the OPE (3.51) has a composite spin-4 primary field A 4 (z). The relative coefficients of descendant fields appearing in the first-order pole are fixed by the formula (1.3). For the second-order pole, as performed before, we compute T (z) {O 2 O 4 ′ } −2 (w) and G(z) {O 2 O 4 ′ } −2 (w) to find the quasi-primary fields in the second-order pole. By subtracting two candidates from the second-order singular terms, we want to check if the following equation holds: It turns out that there is no solution for (3.52). This implies that there should be another primary spin-4 field A 4 (z). Since we already have found two spin-4 primary fields, we expect that the new spin-4 field would be a composite field in terms of known currents (we have determined so far). Otherwise, the spin contents of (1.2) will not be correct. To check this, let us first express A 4 (z) as Then we try to express A 4 (z) in terms of other possible spin-4 fields. It turns out that A 4 (z) can be written as . This field (3.54) containing O 2 O 2 (z) corresponds to the A (4,0) in [7]. As expected it obeys the following OPEs: where A9 2 (z) is a composite primary spin-9 2 field and the superpartner of A 4 (z). It will appear in (3.58) or (3.59) in the next OPE soon. For the first-order pole, the derivative terms are completely determined and one can find the extra quasi-primary fields by the same procedure which we have performed.
The structure constants in (3.51) for general N are given by The relative coefficients of descendant fields are fixed by (1. (z) in the OPE. For the second-order pole, there is a composite primary spin- 9 2 field A9 2 (z) which is the superpartner A 4 (z) in (3.51). It is found by the same method we used to find A 4 (z). The explicit form of A9 In terms of other spin-9 2 fields, it is expressed as where the structure constants are The spin-9 2 field A9 2 (z) obeys the following OPEs: Then, the fields A 4 (z) and A9 2 (z) consist of N = 1 multiplet. For the first-order pole, we compute the following OPEs: Here the descendant fields in the first-order pole contains the first six terms in (3.57). Then we examine higher-order poles with order n > 2 and add extra spin-11 2 quasi-primary fields to the first-order pole, and compute the OPEs with T (z) and G(z) again to check whether the higher-order poles are removed or not. If there are still higher-order poles, we can add another quasi-primary field and compute the OPEs with T (z) and G(z) again. We continue this procedure until the higher-order poles with order n > 2 are completely removed. In this case, removing the higher-order poles was very complicated. When we added the final quasi-primary field, Q11 2 (z), we could successfully remove all higher-order poles. The Q11 2 (z) is given by [7]. The coefficients are determined by the fact that the third-pole of the OPE between T (z) and Q11 2 (w) must vanish. As one can see, Q11 2 (w) is very different from the other quasi-primary fields in the sense that the nonderivative term doesn't contain T (z) or G(z) 8 . The OPEs T (z) Q11 2 (w) and G(z) Q11 2 (w) are put in the Appendix G and Appendix H respectively. After finding Q11 derivative terms and quasi-primary fields with their coefficients in (3.57 It doesn't vanish meaning there is a new primary spin- 11 2 field. It turns out that the new primary spin- 11 2 field cannot be expressed in terms of other spin-11 2 fields meaning it is not a composite field that can be written in terms of known currents. The explicit form of new primary spin- 11 2 , O 11 2 (z), is given by The spin-11 2 current O 11 2 (z) obeys the following OPEs: where O 6 (z) is a primary spin-6 field and the superpartner of O 11 2 (z). The O 6 (w) will appear in (3.72) in the OPE between O 5 2 (z) and O 9 2 (w) soon. To find the N-dependence of the structure constants, we compute the OPE between O 2 (z) and O 9 2 (w) for N = 3, 4, 5, 6 cases. In the previous OPEs, it was enough to compute OPEs up to N = 5 to find the N-dependence of the structure constants. But in this case, we had to compute the OPE between O 2 (z) and O 9 2 (w) up to N = 6 to find the N-dependence of the structure constants completely. To find the structure constants c o , and c go 2 9 2 , we should solve the following equations: To find the structure constants c to = −252c 2 B(29 + 2c)(53 + 2c)(20 + 3c)(21 + 4c)(6 + 5c)(4 + 73c)(c + 6(−1 + N)N) c q 2 9 2 = 4410cB(29 + 2c)(53 + 2c)(20 + 3c)(6 + 5c) 2 (−147 + 182c + 40c 2 ) ≡ 1764(29 + 2c)(53 + 2c)(6 + 5c)(4 + 73c) × (c + 6(−1 + N)N) 2 (20 + 3c)(29 + 2c)(6 + 5c)(c + 6(−1 + N)N), The coefficients in (3.59) are given by The field contents of (3.67) are almost the same as the ones in (3.57 Note that this is equal to (3.62) found before. As done before, we compute the OPE , 5 cases to find the general forms of structure constants. To find the structure constants c o Here the 2nd-order pole contains six terms in (3.67 All results regarding the OPE O 5 (w) Now we compute the OPE between O 5 2 (z) and O 9 2 (w), which is the final and most complicated OPE in this work. From the computation of the OPE for N = 2, we find that (z) is related to the primary field A (6,2) in [7]. The coefficients in (3.71) are determined by the fact that the third-order pole of the OPE between T (z) and Q 6 (w) should vanish. The OPEs T (z) Q 6 (w) and G(z) Q 6 (w) are put in the Appendix G and Appendix H respectively. The O 6 (w) is a spin-6 primary field and the superpartner of spin- 11 2 primary field O 11 2 (z). The O 6 (w) was found by the same procedure used in finding O 11 2 (z). The explicit form of the O 6 (w) is expressed as = 3675cB(29 + 2c)(53 + 2c)(20 + 3c)(21 + 4c) ≡ 735(29 + 2c) 2 (53 + 2c)(20 + 3c)(6 + 5c) 2 (8 + 146c)(c + 6(−1 + N)N) 3 , 4N)). consist of one hundred eighty seven terms, is presented as follows: currents in different coset model were constructed in [19,20]. One can analyze the three-point functions of the spin-6 current with scalars. ]. The explicit OPE is given by (F.6). As described before, among four OPEs between this N = 1 supermultiplet, half of them are quite related to the others because they have common field contents. By rescaling the currents as  5N(−62 + 7N))))), the standard normalizations arise: 2c 9 + · · ·. We have checked this up to N = 5. Then all the previous OPEs can be rewritten in terms of these rescaled currents.

Conclusions and outlook
We have found the first four higher spin supercurrents with spins (2 ′ , 5 2 ), ( 7 2 , 4), (4 ′ , 9 2 ) and ( 11 2 , 6) in (1.2) including the super stress tensor with spins ( 3 2 , 2) in terms of the WZW currents in the coset model (1.1). Some of the OPEs between these supercurrents are determined. In the right hand side of these OPEs, one sees various kinds of quasi-primary (and primary) fields with given spins that can be written in terms of the above higher spin currents 9 .
• So far, the level of the second numerator current of the coset model (1.1) is fixed by 1. It would be interesting to study the higher spin currents for general levels (k, l). Or what happens when the level k is replaced by 2N or (2N + 1) in the coset model? 9 One might ask what is the spin dependence of the maximal degree of the polynomials appearing in the structure constants which can be expressed as a ratio of two polynomials in N . For given higher spin current written in terms of WZW currents, the N dependence arises in many places. That is, the overall factor and the relative coefficient functions between various independent terms. As one calculates the particular OPE between the higher spin currents with spins s 1 and s 2 , these N dpendences occur in each singular term of the OPE. Furthermore, each multiple product of WZW currents of spin s 1 and those of spin s 2 can produce the N dependence also by contracting the group indices during the OPE calculation.
For example, in the OPE of O 2 (z) O 2 (w) given in (3.10), the large N behavior of c 22 can be analyzed as follows. The maximal degrees of each polynomial in the numerator and denominator are given by 4 and 4 respectively from c 22 . The overall factor of O 2 (z) contributes to 1 N 2 and therefore by considering the other O 2 (w), the total contributions in the OPE are given by 1 N 4 . How does one obtain the extra N 4 behavior? The O 2 (z) contains four independent terms. One realizes that the N 4 behavior arises in the OPE between the third term and itself (i.e. K a K b J ab (z) K c K d J cd (w)) and in the OPE between the last term and itself (i.e. J a2N +1 J a2N +1 (z) J b2N +1 J b2N +1 (w)). The fourth-order pole terms of these OPEs behave as N 2 and the relative coefficient functions of third and fourth terms of O 2 (z) behave as N . Therefore, the total contribution is N 2 × N × N = N 4 as above. Note that the contribution from the OPE between the first term and itself in O 2 (z) is given by N 3 (the relative coefficient function in this case is a constant).
For the structure constants we have found in this paper, the maximal degree of polynomial in the numerator is the same as the one in the denominator. Let us denote the maximal degree of polynomial of numerator (or denominator) by deg(N ) and then one realizes that deg(N ) ≤ s 1 + s 2 . The c dependent coefficients appearing in the numerator and denominator can be determined by deg(N ) + 1 linear equations which can be obtained from the expressions for lower N = 2, 3, · · · , deg(N ) + 2. Most of the structure constants have their factorized forms and therefore, we do not need all the above deg(N ) + 1 linear equations to determine the c-dependent coefficients. Of course, as the spins s 1 and s 2 increase, the deg(N ) becomes large and it will take too much time (by package) to obtain the complete OPEs for low several N values. The spin s 3 of higher spin current appearing in the right hand side of above OPE is less than s 1 + s 2 : s 3 < s 1 + s 2 . There is no definite relation between the s 3 and deg(N ). In som examples, deg(N ) is greater than s 3 but in other examples, deg(N ) is less than or equal to s 3 . Therefore, the final c-dependent coefficient functions (i.e. structure constants of the OPEs) for general N can be obtained.
• According to recent work in [21], the large N = 4 minimal model holography is an interesting subject. For example, there exists a particular N = 4 coset theory related to the orthogonal group as follows: where W is a Wolf space. Simple computation for the central charge in this model leads to c = 6(k+1)(N +1) (k+N +2) which is exactly the same as the central charge studied in [21]. The immediate step is how to construct the large N = 4 superconformal algebra in this particular coset theory. For the Wolf space itself, the subgroup of SO(N + 4) is realized by SO(N) × SO(4) = SO(N) × SU(2) × SU(2).
One can study this coset theory for fixed N in order to see the structure of an extended version of large N = 4 superconformal algebra.
• Furthermore, one of the Kazama-Suzuki models has the following coset model SO(N +2) where the central charge is given by c = 3N k (N +k) . It would be interesting to find the higher spin currents, along the line of [22,23]. First of all, the N = 2 superconformal algebra should be realized in this coset model from the N = 2 WZW currents with constraints. After this is done, then the extension of the N = 2 superconformal algebra can be obtained by constructing the higher spin currents with spins greater than 2.
The two quasi-primary fields in (G.1) and (G.