The ${\cal N}=4$ Supersymmetric Linear $W_{\infty}[\lambda]$ Algebra

From the recently known ${\cal N}=2$ supersymmetric linear $W_{\infty}^{K,K}[\lambda]$ algebra where $K$ is the dimension of fundamental (or antifundamental) representation of bifundamental $\beta \, \gamma$ and $b \, c$ ghost system, we determine its ${\cal N}=4$ supersymmetric enhancement at $K=2$. We construct the ${\cal N}=4$ stress energy tensor, the first ${\cal N}=4$ multiplet and their operator product expansions (OPEs) in terms of above bifundamentals. We show that the OPEs between the first ${\cal N}=4$ multiplet and itself are the same as the corresponding ones in the ${\cal N}=4$ coset $\frac{SU(N+2)}{SU(N)}$ model under the large $(N,k)$ 't Hooft-like limit with fixed $\lambda_{co} \equiv \frac{(N+1)}{(k+N+2)}$, up to two central terms. The two parameters are related to each other $\lambda =\frac{1}{2}\, \lambda_{co}$. We also provide other OPEs by considering the second, the third and the fourth ${\cal N}=4$ multiplets in the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$ algebra.


Introduction
The free field construction in two dimensional conformal field theory is useful to study the extension of the conformal symmetries in string theory. Because their operator product expansions (OPEs) take the simple form in the sense that the right hand side of OPEs does not contain the fields, contrary to the affine Kac-Moody algebra, it is straightforward to determine the conserved currents of any (conformal) weights in terms of the quadratic free fields with multiple derivatives. In order to describe the supersymmetric theory, the fermionic free field is necessary to describe the symmetries as well as the bosonic free field. Depending on the weights of the bosonic and fermionic fields, the weights of the currents we can make by using them are determined naturally from a simple counting of weights. The central charge of the Virasoro algebra consisting of the stress energy tensor of weight-2 is fixed by the number of (bosonic and fermionic) free fields. Usually, the bosonic field has the weight-1 (or zero) while the fermionic field has the weight- 1 2 . More generally, the above weights of the bosonic and fermionic fields can be deformed by a parameter λ [1]. Although the weights of each bosonic and fermionic field depend on this λ parameter explicitly, due to the plus and minus signs in the coefficients in front of λ of the weights, the weights of particular composite combinations of these free fields do not contain the parameter λ. Then we can construct the currents of integer (or half integer) weights in terms of free fields as mentioned before by considering that the weights of each term should not depend on the λ. Of course, the λ-dependence in the coefficients in front of free fields in the expression of the currents occurs in very nontrivial way [2,3]. This is a new feature because the structure constants of the resulting algebra contain the λ dependence explicitly, compared to the ones in the previous paragraph.
So far, we have two bosonic and two fermionic free fields. There exist two fundamental OPEs between them. We can introduce the multiple bosonic and fermionic fields which transform as bifundamentals. Because they are independent fields and the multiple defining OPEs satisfy independently, all the previous analysis can be generalized to describe the symmetries easily. For example, the central charge of the Virasoro algebra is simply a sum over each contribution from bosonic and fermionic free fields. For each current of weight-h, there exist multicomponent generators. The corresponding W ∞ algebras (without any deformation parameter λ) are obtained in [4,5,6]. By construction, because there are many fermionic currents, there is more room for the supersymmetric theory we would like to obtain. Then it is natural to consider the multiple bosonic and fermionic free fields together with the deformation of λ. Recently, in [7], the N = 2 supersymmetric linear W K,K ∞ [λ] algebra is obtained by analyzing the currents from the multiple bosonic and fermionic free fields with derivatives. Here K is the dimension of fundamental representation of above bifundamentals. The number of bosonic currents of each weight (h = 1, 2, · · ·) is given by 2K 2 which is equal to the number of fermionic currents of each weight (h = 3 2 , 5 2 , · · ·). The factor 2 appears because we are considering the complex free fields. Among 2K 2 -fermionic currents, two of them play the role of N = 2 supersymmetry generators. There are also K 2 -fermionic currents of weight- 1 2 and this fact will affect the structure of the N = 4 superconformal algebra 1 . In this paper, we would like to construct the N = 4 supersymmetric linear W 2,2 ∞ [λ] algebra by focusing on the K = 2 case which is very special in the sense that only this K = 2 will provide the supersymmetric theory we want to obtain. Then among eight fermionic currents of weight- 3 2 , the half of them play the role of N = 4 supersymmetry generators. The remaining half of them belong to the first N = 4 multiplet. Moreover, the lowest fermionic currents of weight- 1 2 can join the generators of the N = 4 superconformal algebra. For the weight-1 currents, the seven of them play the role of the bosonic generators of the N = 4 superconformal algebra and the remaining one will appear in the lowest operator in the first N = 4 multiplet. For the weight-2, one of them is given by the stress energy tensor, six of them will appear in the generators of the first N = 4 multiplet and the remaining one will arise in the lowest operator in the second N = 4 multiplet. For the weights greater than 2, we can analyze similarly and they can be placed into the corresponding N = 4 multiplets appropriately, according to SO (4) indices of N = 4 superspace.
We would like to determine the explicit algebra how the above analysis on the weight contents fits in the N = 4 supersymmetric linear W ∞ [λ] algebra 2 .
In this paper, we determine the N = 4 stress energy tensor, the first, the second, the third and the fourth N = 4 multiplets in the N = 4 supersymmetric W ∞ [λ] algebra, by using the β γ and b c ghost systems. We calculate the various OPEs between them, where the sum of two (super) weights appearing on the left hand side is less than or equal to 4 (h 1 + h 2 ≤ 4), in N = 4 superspace. As in the abstract, the case of h 1 = 1 = h 2 reproduces the corresponding one [10,11] in the N = 4 coset model under the large (N, k) 't Hooft-like limit.
In section 2, we review the β γ and b c ghost systems and the bosonic and fermionic currents can be written in terms of these fields.
In section 3, we determine the N = 4 stress energy tensor, the first N = 4 multiplet and their OPEs explicitly.
In section 4, we summarize what we have obtained in this paper and future directions are 1 There are also similar constructions in [8,9]. 2 The terminology of W ∞ [λ] algebra is used here instead of using the previous terminology of W 2,2 ∞ [λ] algebra for simplicity.

described.
In Appendices, some of the details in section 3 are presented explicitly. We are heavily using the Thielemans package [12] with the help of a mathematica [13].

The quadratic bosonic and fermionic operators
We can construct the bosonic and fermionic operators (or currents) by taking the quadratic expressions of above β γ and b c ghost systems in the presence of various holomorphic derivatives as follows: [8,7] The first two operators of weight s are bosonic and the last two operators of weight (s − 1 2 ) are fermionic. Each term on the right hand sides has the summation over the indices l andl of SU(N). Each operator has four components, 11, 12, 21 and 22 in the indicesā and b. Each coefficient on the right hand sides depends on the weight s (or (s − 1 2 )) and the λ. They can be summarized by [2,3] The parentheses in (2.3) stand for the binomial coefficients and the (a) n symbols stand for the rising Pochhammer symbol (a) n ≡ a(a + 1) · · · (a + n − 1). There are nontrivial relations between these coefficients [2, 3].

The λ-dependent currents
We can split the above bosonic operators into the one written in terms of bosonic fields and the other written in terms of fermionic fields by simple linear combinations. For the fermionic operators, we also split them in terms of the one having only one kind of fermion fields and the other having only the other kind of fermion fields. Then we obtain the following operators with the explicit bifundamental indices [7] W λ,āb The first two operators of weight h are bosonic and the last two operators of weight (h+ 1 2 ) are fermionic 3 . The overall coefficients do not depend on the λ. We list the explicit expressions for low weights by substituting (2.3) and (2.5) into (2.4) as follows [7]: The normalizations are given by [14] n Then the q dependence in (2.4) disappears.
Note that the right hand sides of (3.4) are proportional to the expressions of Φ (1),ij 1 at λ = 0 when we replace the weight-2 in the Wā b B,2 and Wā b F,2 [10] with the weight-1 by generalizing to the λ dependent ones. We can check that the first order pole of the OPE G i (z) G j (w) provides the correct quasiprimary operator L(w) and the corresponding descendants of (3.4). 1 2 do not contain the λ from (2.2) and (2.4).

The λ-independent weight-1 quasiprimary operator
For the final weight-1 operator, we can use the defining equation for the first order pole of the OPE G i (z) Γ j (w) which is equal to − ε ijkl T kl + i δ ij U (w). It turns out that which does not contain the λ. Therefore, the eight independent weight-1 operators from Wā b B,2 and Wā b F,2 are given by (3.4) and (3.6). The remaining one will be given in next subsection of the first N = 4 multiplet.
In next subsection, we will describe whether the above five kinds of operators will produce the known N = 4 superconformal algebra or not.

3.2
The OPEs between the N = 4 stress energy tensor and itself We calculate the OPEs between the five operators in the N = 4 stress energy tensor and the weight-1 operator in that multiplet.

3.2.1
The OPE between the weight-1, 1 2 , 1 operators and the weight-1 operator The last two are the standard results [15,11] while the first one is rather trivial result. This is due to the fact that the expression for the U in (3.6)has the same relative coefficients. The standard result for the first one leads to the nontrivial second order pole, which is given by a central term. We expect that the OPE between L(z) U(w) contains the third order pole because the OPE ∂ U(z) U(w) has no singular term.

The OPE between the weight-3 2 operators and the weight-1 operator
Similarly, by using (3.3), (3.6) and previous defining equations, we obtain the following OPE The weight-1 operator U plays the role of keeping the structure of the weight-3 2 operator on the left hand side with the weight reduced to 1 2 . The relative coefficient for the descendant can be seen from the standard conformal field theory analysis 5 .

The OPE between the weight-2 operator and the weight-1 operator
Let us consider the final nontrivial OPE with (3.1) and (3.6). It turns out that there exists Compared to the standard result [15,11], the above OPE contains the central term, as mentioned before. This is due to the fact that the expression of (3.6) has particular relative coefficients 6 . 5 There is due to one of the relations in (3.7). This will be used in the N = 4 superspace description. 6 We have ∂U (w) + · · · , (3.11)

The N = 4 supersymmetric OPE in the N = 4 superspace
We can put the operators of N = 4 superconformal algebra found in previous section into a super field in the N = 4 superspace 7 . Then we have the N = 4 stress energy tensor [15,11] where the lowest component has the following relation −∂ ∆ ≡ U. We will use the operator U rather than the operator ∆. The precise relations between the components and its superfields in (3.12) at vanishing fermionic coordinates can be summarized by [15,11] U ↔ ∂ J, Due to the extra terms in the fourth and fifth elements of the N = 4 stress energy tensor, there are extra terms in the corresponding expressions of (3.13).
Then we can write down the previous equations (3.7), (3.8) (or (3.9)) and (3.10) (or (3.11)) including other OPEs in the component approach in terms of the following single OPE in the N = 4 superspace (3.14) Compared to the standard expression [15,11], there is no log(z 12 ) term 8 .
which will be used later. 7 The coordinates of N = 4 superspace can be described as (Z, Z) where Z = (z, θ i ), Z = (z,θ i ) and the SO(4)-vector index i runs over i = 1, · · · , 4. The left covariant spinor derivative is given by [15,11]. 8 We have the corresponding OPE In Appendix A, all the component OPEs are summarized explicitly 9 . It is straightforward to obtain these OPEs from (3.14) by acting the various super derivatives with the relations (3.13).

The construction of the first
The λ-dependent weight-1 primary operator Let us start with the final weight-1 primary operator of the N = 2 superconformal algebra [7,17] Φ (1) The field contents of (3.16) are the same as the one in (3.6). At λ = 0, the only first two terms in (3.16) contribute to the final expression and reproduces the one in [10]. Compared to the previous construction on the weight-1 operator, the λ-dependent coefficients appear in the above.

The λ-dependent weight-3 2 primary operators
From the defining equation [15,18,11] of we can determine the following primary (under the stress energy tensor (3.1)) operators of weight-3 Compared to the λ = 0 case in [10], the generalization of the fermionic operators to the nonzero λ case in (2.4) provides the exact relative coefficients in (3.18). In other words, for the expressions in [10] at λ = 0, a simple generalization of (2.4) leads to the above result in (3.18). This is also true for other remaining operators of weights-2, 5 2 , 3.

3.3.3
The λ-dependent weight-2 primary operators By using the following defining equation [11] G we obtain the weight-2 primary operators, by taking two different indices, as follows: We observe, as described before, that by taking the corresponding expressions for the weight-1 operators at λ = 0 and replacing them with the ones in (2.4), the above results can be obtained. So far, we have obtained the seven weight-2 operators consisting of (3.1) and (3.20) and the the remaining one will appear in the lowest component of the second N = 4 multiplet.

3.3.4
The λ-dependent weight-5 2 quasiprimary operators From the defining equation [11] of we can determine the following weight-5 2 quasiprimary operators Note that by starting with (3.18) with minus signs in Q λ,āb 3 2 and increasing the weights by one, we reproduce the above results (3.22). For the case of λ = 0 in [10], the weight-5 2 operators are primary but at nonzero λ, the above operators (3.22) are quasiprimary under the stress energy tensor (3.1), although we are using the same notation 10 . We expect that the half of other weight- 5 2 operators will appear in the second N = 4 multiplet.

3.3.5
The λ-dependent weight-3 quasiprimary operator Finally, by using the following defining equation [11] G we obtain the weight-3 quasiprimary operator, by taking two equal indices, We observe that by increasing the weight by one from the stress energy tensor (3.1), the above expression can be seen with the overall factor. Compared to the λ = 0 case in [10] where the corresponding operator is primary, the above operator is quasiprimary 11 . The remaining seven other weight-3 operators will appear in next N = 4 multiplets. Six of them appear in the second N = 4 multiplet and one of them appears in the third N = 4 multiplet. 10 The operators Φ (1),i 3 2 , which are the component of N = 4 superfield later, are not quasiprimary. See also Appendix B. 11 The operator Φ 2 is not quasiprimary and see also Appendix B.

3.4
The OPEs between the N = 4 stress energy tensor and the first N = 4 multiplet We calculate the OPEs between the five kinds of operators in the N = 4 stress energy tensor and the lowest weight-1 operator in the first N = 4 multiplet.

The OPE between the weight-1 operator and the weight-1 operator
From the explicit expressions in (3.6) and (3.16), the following OPE can be obtained (3.25) Compared to the standard result [11] which is trivial, the above OPE has singular term on the right hand side. This is due to the fact that this weight-1 operator has the particular coefficients.

The OPE between the weight-1 2 operators and the weight-1 operator
Similarly, we obtain the following OPE from (3.5) and the previous weight-1 operator This implies that the weight-1 operator preserve the structure of the weight-1 2 operator on the left hand side and this is new, compared to the standard result [11].

The OPE between the weight-1 operators and the weight-1 operator
For the weight-1 operator (3.4), we obtain the following trivial result (3.27)

3.4.4
The OPE between the weight-3 2 operators and the weight-1 operator By using (3.3) and (3.16), the following OPE is satisfied Under the action of the weight-1 operator, the numerical coefficients appearing in the weight-3 2 operators are shifted to the ones in the weight-3 2 operators appearing on the right hand side 12 . 12 We have the OPE

The OPE between the weight-2 operator and the weight-1 operator
Finally, the last fundamental OPE from the stress energy tensor (3.1) can be summarized by This implies that the weight-1 operator is primary 13 .
3.4.6 The N = 4 supersymmetric OPE in the N = 4 superspace As before, we write down each component operator of the first N = 4 multiplet in the N = 4 superspace as follows [15,18,11]: The precise relations between the components and its superfields in (3.32) can be described by [15,11], similar to (3.13), In other words, by taking the fermionic coordinates on the right hand sides to zero, we obtain the corresponding operators on the left hand sides.
This implies that the first N = 4 multiplet is not a primary operator under the N = 4 supersymmetry because there are the first four terms in (3.34).
In Appendix B, we present all the component OPEs explicitly. As before, these can be checked by using the super derivatives in [11].

3.5
The OPEs between the first N = 4 multiplet and itself We calculate the OPEs between the five kinds of operators in the first N = 4 multiplet and the lowest weight-1 operator in that multiplet.

The OPE between the weight-1 operator and itself
By using the equations, (3.16), (2.6), (2.2) and (2.1), we obtain the following OPE (3.35) Note the presence of the factor (1 − 4λ) in the above.

The OPE between the weight-3 2 operators and the weight-1 operator
With the explicit expressions (3.18) and the previous defining relations, we obtain the following OPEs This implies that the role of the weight-1 operator Φ 0 (w) in this OPE changes the signs of Q λ,āb 3 2 . This leads to the first order pole on the right hand side of the above OPE.

The OPE between the weight-2 operators and the weight-1 operator
With the help of (3.20), we determine the following OPEs In this case, the role of Φ 0 (w) in this OPE decreases the weight of the weight-2 operator on the left hand side by one and it turns out that the second order pole of above OPE is a linear combination of (3.4). We have mentioned that there are some similarities in the weight-2 operator Φ (1),ij 1 and the weight-1 operator T ij . Furthermore, the relative coefficients 1 and 1 between the second and first order poles on the right hand side can be understood from the property of the standard conformal field theory analysis based on the weights of Φ From the expressions of (3.22), we determine the following OPEs Note that the weight-5 2 operator on the left hand side is not a quasiprimary operator, as mentioned before. Therefore, the coefficients of the descendants in the above OPE are not known in general. It turns out that according to the realization of the β γ and b c ghost system, we obtain the above result. If we use the quasiprimary weight- 5 2 operatorΦ (1),i 3 2 , then the coefficient of the G i in the second order pole is given by 8 3 (1 − 4λ) and others remain the same. The contributions from the extra terms in theΦ . After subtracting the descendants in the first order pole, we are left with a new quasiprimary operator which cannot be written in terms of the previously known operators. It turns out that there are Φ (2),1 We realize that this looks similar to the ones in (3.18) in the sense that we obtain the above expressions by increasing the weight of the right hand sides of (3.18) by one and multiplying the overall factor −2. At this moment, it is not clear how the numerical factor 1 2 in the coefficient 1 2 of Φ (2),i 1 2 in the first order pole appears.
3.5.5 The OPE between the weight-3 operator and the weight-1 operator By using the expression in (3.24), we obtain Similarly, for the quasiprimary weight-3 operatorΦ It turns out that we obtain the lowest operator in the second N = 4 multiplet shifted by the stress energy tensor The structure of the right hand side looks similar to the one in (3.16) and we obtain the above expression by increasing the weight of the right hand side of (3.16) by one and multiplying the overall factor −2. Also it is not clear how the numerical factor 2 in front of the second terms in the second order pole appears 14 .  14 We can write down the weight-2 operator from (3.41) as Φ The λ dependence appears as (1 − 4λ) or λ (1 − 2λ) except the first central term of (3.43). We have checked that the above OPE, except the two central terms appearing in the first two lines of (3.43), is the same as the one [10] under the large (N, k) limit.
In Appendix C, we write down all the component OPEs explicitly for convenience. Equivalently, all the OPEs in (C.1) can be obtained from (3.43) by acting the various super deriva-tives on both sides and putting the fermionic coordinates to zero. In Appendix D, the five fundamental OPEs in the N = 4 coset model under the large (N, k) limit are given explicitly in (D.1). We observe that the previous OPEs (3.35), (3.36), (3.37), (3.38) and (3.40) are identical to the ones in (D.1) together with λ = 1 2 λ co in the footnote 4. Furthermore, other OPEs appearing in (C.1) are identified with the ones in [10] we do not present in this paper.

3.6
The OPEs between the other N = 4 multiplets From the defining equation [15,18,11] of which is obtained from the relation (3.17) by changing the weight properly, we can determine the quasiprimary operators of weight-5 2 appearing in (3.39) by using (3.44). By using the following defining equation [11] G  .
From the defining equation [11] of (3.47) which can be obtained from (3.21), we can determine the following weight-7 2 quasiprimary operators by taking the appropriate indices in (3.47) . (3.48) Finally, by using the following defining equation [11] G i (z) Φ (2),j which is obtained from (3.23), we obtain the weight-4 quasiprimary operator, by taking two equal indices in (3.49),  Moreover, their algebras between these generators are determined and in particular, the OPEs between the first N = 4 mutiplet and itself are equivalent to the corresponding ones in the N = 4 coset model under the large (N, k) limit. Contrary to the findings in [19], the modes of the currents in the present results are not restricted to the wedges but can have any integers or half integers because our construction is based on the OPEs between the currents.
So far, we have considered the OPEs between the N = 4 stress energy tensor and the first N = 4 multiplet (and other OPEs in Appendices E, · · · , J). Then it is natural to ask what are the OPEs between the h 1 -th N = 4 multiplet and the h 2 -th N = 4 multiplet for any weights h 1 and h 2 . In the analysis of (G.2), we can figure out the explicit form for the five kinds of currents for general weight-h. The lowest component can be obtained easily up to the overall normalization. The remaining components can be also determined with the weight dependent overall factors. Then the question is how we can write down the OPEs between the currents appearing in (2.4) for each component in terms of the λ dependent structure constants introduced in [19]. It would be interesting to rewrite all the structure constants obtained in this paper in terms of previously known ones presented in [7]. This will give us some hints to figure out their behaviors for generic weights 15 . 15 For example, the structure constant appearing in W λ,āb F,2 δ bā (w) of the OPE between W λ,āb F,4 δ bā (z) and W λ,āb F,4 δ bā (w) is given by 2048 5 (λ−1) (λ+1) (2λ−3) (2λ+3) around the equation (3.18) in [7]. This λ dependent function is related to p 4,4 F,4 (m, n, λ) appearing in that paper. By realizing that we can extract W λ,āb F,4 δ bā from the present context and we have W λ,āb F,4 δ bā = 1 28 (3 + 2λ) Φ 0 , we can check that the sixth order pole in the OPE between these three terms and itself reproduces the above structure Because the second N = 4 multiplet from the free field approach is not directly related to the corresponding N = 4 multiplet from the coset fields in [10] (for example, the OPE between the Φ (1) 0 and Φ (2) 0 in the former does not vanish while that in the latter does vanish), it would be interesting to obtain the correct second N = 4 multiplet in the coset model at finite (N, k) as a first step. Note that according to the free field approach in this paper, the currents (3.42), (3.39), (3.46), (3.48) and (3.50) are the quasiprimary operators under the stress energy tensor. We need to find out the correct basis where the corresponding currents in the N = 4 coset model should reflect this quasiprimary condition at least by calculating all the nonlinear terms. After that, we also expect that under the large (N, k) limit, for example, the OPEs between the first N = 4 multiplet and the second N = 4 multiplet in the coset model will produce the ones on Appendix F .
In the context of celestial holography [20], we have seen that the wedge subalgebra of w 1+∞ algebra [21] provides the symmetries on the celestial sphere [22]. See also [23]. Moreover, the analysis for the N = 1 supersymmetric w 1+∞ algebra is obtained in [24,25]. In the present context, the above w 1+∞ algebra is related to the OPEs between the currents W λ,āb F,h δ bā with the structure constants p h 1 ,h 2 F,h (m, n, λ) described in the footnote 15. In the context of N = 4 supersymmetric linear W ∞ [λ] algebra, the current of weight-h are made of 1) the lowest current in the h-th N = 4 multiplet, 2) the middle current in the (h − 1)-th N = 4 multiplet and 3) the highest current (and the lowest current with two derivatives) in the (h − 2)-th N = 4 multiplet. It would be interesting to observe whether the corresponding supersymmetric Einstein-Yang-Mills theory at nonzero deformation parameters λ (or q) reveals the OPEs we have obtained in this paper or not.
In [2,3], the explicit representation of the corresponding algebra is given by the differential operators in terms of commuting parameters Λ ± (z) and anticommuting parameters Θ ± (z) 16 . Under the symmetry generated by the currents in this paper, the transformation of any fields (or operators) is given by the following contour integrals over z with the OPEs between the currents and the fields (along the lines of [26]) constant. Furthermore, by using the weight-4 current W λ,āb B,4 δ bā = 1 (1) 0 , we obtain the sixth order pole in the OPE between them, which is equal to 512 (λ − 1) λ (2λ − 3) (2λ + 1). This structure constant appears in the equation (B.2) of [7] and is related to p 4,3 B,4 (m, n, λ).
In these examples, the bifundamental indices are contracted with each other. However, we should also obtain the OPEs between the currents with free bifundamental indices for generic weights. 16 We thank the referee for pointing the questions raised in the remaining paragraphs out.
where the contour C w surrounds the point z and there are no summations over the indicesā and b on the right hand sides of (4.1). We can describe the corresponding (anti)commutator relations between the 'charges' and the fields. By using the result of the following OPE and substituting this (4.2) into the (4.1), we obtain This implies that we realize that there exist the corresponding linear differential operators appearing inside the bracket in (4.3). Therefore, the nontrivial differential operators occur only when the first elementā of the currents V (s)+ λ,āb (z) and the second element c of the operators βm c (w) are equal to each other.
Similarly, we act the above currents on the bj c (w). From the result of we determine the following transformation with (4.4) We observe that there exist the corresponding linear differential operators appearing inside the bracket in (4.5).
Due to the multiple derivatives of βj b (w) in the currents V (s)+ λ,āb (z), the next OPE is rather complicated and it turns out that (4.6) Note that there is a summation over t and its maximum number is given by (i + 1). There is also a summation over i.
Then we obtain the following result by using the relation (4.6) There exist the corresponding linear differential operators with the double summations appearing inside the bracket in (4.7).
Again, due to the multiple derivatives of bj b (w) in the currents V (s)+ λ,āb (z), the next OPE is complicated and it turns out that (4.8) The corresponding transformation, with the help of (4.8), can be written as The corresponding linear differential operators with the double summations appearing inside the bracket in (4.9) occur. Now we can consider the symmetry generated by fermionic currents and they can be described as follows: Therefore, we have the transformations of the β γ and b c ghost systems under the bosonic and fermionic currents, summarized by (4.3), (4.5), (4.7), (4.9) and (4.10). Furthermore, for the remaining bosonic and fermionic currents, we can calculate the corresponding OPEs with the β γ and b c ghost systems and we summarize the following results It is straightforward to obtain the corresponding transformations of the β γ and b c ghost systems for the N = 4 currents studied in this paper, by taking the linear combinations between the above results summarized in (4.3), (4.5), (4.7), (4.9), (4.10) and (4.11).

A The OPEs between the N = 4 stress energy tensor and itself in the component approach
For the previous result (3.14), the component results can be summarized by Compared to the large N = 4 superconformal algebra [15,11], there are two additional central terms in the fifth and eighth of (A.1) 17 . Moreover, there are trivial OPEs Γ i (z) Γ j (w) and U(z) U(w). Finally, the central term of the OPE T ij (z) T kl (w) which is proportional to (δ ik δ jl − δ il δ jk ) does not appear.
B The OPEs between the N = 4 stress energy tensor and the first N = 4 multiplet in the component approach We present the complete OPEs corresponding to (3.34) as follows: Compared to the N = 4 primary condition [11] of the first N = 4 multiplet, there are additional terms in (B.1): either central terms or the field contents of the N = 4 stress energy tensor 18 . In other words, the first N = 4 multiplet is not N = 4 primary.

C
The OPEs between the first N = 4 multiplet and itself in the component approach As before, we list all the component results corresponding to (3.43) as follows: 2 (z) Φ (1) 0 (w) = 1 (z − w) 2 2i (1 − 2λ co ) T ij + i ε ijkl T kl (w) It is straightforward to express the OPE like as in (3.43) in N = 4 superspace.

E
The OPEs between the N = 4 stress energy tensor and the second N = 4 multiplet On the right hand sides of (F.1), there are the N = 4 stress energy tensor, the first N = 4 multiplet, the third N = 4 multiplet as well as their descendants. The third N = 4 multiplet can be summarized by Φ