The Structure of

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

Changhyun Ahn 111On leave from the Department of Physics, Kyungpook National
University, Taegu 41566, Korea and address during Sept. 01, 2022-Feb. 28,
2023: Department of Physics, New York University, 726 Broadway, New York, NY
10003, USA

Department of Physics, Kyungpook National University, Taegu 41566, Korea

Department of Physics, New York University, 726 Broadway, New York, NY 10003,
USA

<EMAIL_ADDRESS>
Abstract

For the vanishing deformation parameter $\lambda$, the full structure of the
(anti)commutator relations in the ${\cal N}=4$ supersymmetric linear
$W_{\infty}[\lambda=0]$ algebra is obtained for arbitrary weights $h_{1}$ and
$h_{2}$ of the currents appearing on the left hand sides in these
(anti)commutators. The $w_{1+\infty}$ algebra can be seen from this by taking
the vanishing limit of other deformation parameter $q$ with the proper
contractions of the currents. For the nonzero $\lambda$, the complete
structure of the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$
algebra is determined for the arbitrary weight $h_{1}$ together with the
constraint $h_{1}-3\leq h_{2}\leq h_{1}+1$. The additional structures on the
right hand sides in the (anti)commutators, compared to the above $\lambda=0$
case, arise for the arbitrary weights $h_{1}$ and $h_{2}$ where the weight
$h_{2}$ is outside of above region.

On the occasion of my thirtieth Ph.D. anniversary

###### Contents

  1. 1  Introduction and Outlook
  2. 2  The ${\cal N}=4$ multiplet
    1. 2.1 Review
    2. 2.2 The ${\cal N}=4$ multiplet
      1. 2.2.1 The lowest component
      2. 2.2.2 The second components
      3. 2.2.3 The third components
      4. 2.2.4 The fourth components
      5. 2.2.5 The last component
  3. 3  The ${\cal N}=4$ supersymmetric linear $W_{\infty}^{2,2}$ algebra between the adjoints and the bifundamentals under the $U(2)\times U(2)$ symmetry
    1. 3.1  The (anti)commutator relations between the nonsinglet currents
      1. 3.1.1 The commutator relation with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$
      2. 3.1.2 The second commutator relation with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$
      3. 3.1.3 Other commutator relations with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
      4. 3.1.4 The final anticommutator relation with $h_{1}=h_{2}$ for nonzero $\lambda$
    2. 3.2 The (anti)commutator relations between the nonsinglet currents in explicit forms with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$
  4. 4  The ${\cal N}=4$ supersymmetric linear $W_{\infty}^{2,2}$ algebra between the ${\cal N}=4$ multiplets
    1. 4.1 The commutator relation between the lowest components with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$
    2. 4.2 The commutator relation between the lowest component and the second component with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    3. 4.3 The commutator relation between the lowest component and the third component with $h_{1}=h_{2},h_{2}+1,h_{2}+2$ for nonzero $\lambda$
    4. 4.4 The commutator relation between the lowest component and the fourth component with $h_{1}=h_{2}+1,h_{2}+2$ for nonzero $\lambda$
    5. 4.5 The commutator relation between the lowest component and the last component with $h_{1}=h_{2}+1,h_{2}+2,h_{2}+3$ for nonzero $\lambda$
  5. A The structure constants
  6. B The other central terms
  7. C The OPEs between the nonsinglet currents
    1. C.1  The OPE between the bosonic currents
      1. C.1.1  The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and itself
      2. C.1.2  The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and $W_{F,4}^{\lambda,\hat{B}=2}$
      3. C.1.3 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and itself
      4. C.1.4 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and $W_{B,4}^{\lambda,\hat{B}=2}$
    2. C.2 The OPE between the bosonic currents and the fermionic currents
      1. C.2.1 The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and $Q_{\frac{7}{2}}^{\lambda,\hat{B}=1}$
      2. C.2.2  The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and $Q_{\frac{7}{2}}^{\lambda,\hat{B}=2}$
      3. C.2.3 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and $Q_{\frac{7}{2}}^{\lambda,\hat{B}=1}$
      4. C.2.4 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and $Q_{\frac{7}{2}}^{\lambda,\hat{B}=2}$
      5. C.2.5 The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=1}$
      6. C.2.6 The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=2}$
      7. C.2.7 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=1}$
      8. C.2.8 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=2}$
    3. C.3 The OPE between the fermionic currents
      1. C.3.1 The OPE between $Q_{\frac{7}{2}}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=1}$
      2. C.3.2 The OPE between $Q_{\frac{7}{2}}^{\lambda,\hat{A}=1}$ and $\bar{Q}_{\frac{7}{2}}^{\lambda,\hat{B}=2}$
  8. D The other central terms in the OPEs between the fermionic currents
  9. E The other (anti)commutator relations
    1. E.1  The commutators between the currents consisting of $\beta\,\gamma$ system with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$
    2. E.2  The commutators between the currents consisting of $b\,c$ system and the currents consisting of $\gamma\,b$ system with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    3. E.3  The commutators between the currents consisting of $\beta\,\gamma$ system and the currents consisting of $\gamma\,b$ system with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    4. E.4 The commutators between the currents consisting of $b\,c$ system and the currents consisting of $\beta\,c$ system with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    5. E.5 The commutators between the currents consisting of $\beta\,\gamma$ system and the currents consisting of $\beta\,c$ system with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    6. E.6 The anticommutators between the currents consisting of $\gamma\,b$ system and the currents consisting of $\beta\,c$ system with $h_{1}=h_{2}$ for nonzero $\lambda$
  10. F Some relations which can be used in section $4$
  11. G The remaining (anti)commutator relations between the ${\cal N}=4$ multiplets
    1. G.1 The anticommutator relation between the second components with $h_{1}=h_{2}$ for nonzero $\lambda$
    2. G.2 The commutator relation between the second component and the third component with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    3. G.3 The anticommutator relation between the second component and the fourth component with $h_{1}=h_{2}+1$ for nonzero $\lambda$
    4. G.4 The commutator relation between the second component and the last component with $h_{1}=h_{2}+1,h_{2}+2$ for nonzero $\lambda$
    5. G.5 The commutator between the third components with $h_{1}=h_{2}-1,h_{2},h_{2}+1$ for nonzero $\lambda$
    6. G.6 The commutator relation between the third component and the fourth component with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    7. G.7 The commutator relation between the third component and the last component with $h_{1}=h_{2},h_{2}+1,h_{2}+2$ for nonzero $\lambda$
    8. G.8 The anticommutator relation between the fourth components with $h_{1}=h_{2}$ for nonzero $\lambda$
    9. G.9 The commutator relation between the fourth component and the last component with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$
    10. G.10 The commutator relation between the last components with $h_{1}=h_{2}-1,h_{2},h_{2}+1$ for nonzero $\lambda$

##  1  Introduction and Outlook

In the description of [1, 2], the conformal weights of the free bosonic and
fermionic operators in the two dimensional conformal field theory do depend on
the deformation parameter $\lambda$. See also the relevant work [3] for the
role of this $\lambda$ in the similar two dimensional model. The bosonic and
fermionic currents made from the above free fields quadratically by including
the multiple derivatives have integer or half integer weights. The algebra
from these currents has the $\lambda$ dependent structure constants on the
right hand sides of the (anti)commutator relations because the relative
coefficients between the free fields in the expression of the currents reveal
the $\lambda$ dependence in nontrivial way. By construction in [1, 2], it is a
new feature, compared to the previous construction by using free fields (for
example, [4, 5, 6]), that there exist the bosonic current of weight-$1$ and
the fermionic current of weight-$\frac{1}{2}$. The multiple number of free
bosonic and fermionic operators can be introduced. Then it is straightforward
to write down the corresponding algebra because the defining operator product
expansions(OPEs) between these multiple free fields satisfy independently. See
also the relevant work in [7] without a deformation parameter we mentioned
above.

The structure of the ${\cal N}=2$ supersymmetric linear
$W_{\infty}^{K,K}[\lambda]$ algebra where $K$ is the number of complex bosons
or the number of complex fermions is found in [8]. The structure constants for
vanishing deformation parameter are generalized to the ones for nonzero
$\lambda$. However, it turns out that there are some constraints in the
weights for the currents on the left hand sides of the algebra according to
the result of this paper. Of course, when we go from the algebra i) at
$\lambda=0$ to the algebra ii) at $\lambda\neq 0$, once the corresponding
structure constants in the latter which lead to the ones in the former when we
take $\lambda\rightarrow 0$ are determined, then definitely we expect to have
the generalized structure constants for nonzero $\lambda$ in the algebra ii)
via above transition. This is the simplest deformed algebra.

On the other hand, suppose that there are some “additional” structure
constants having the factor $\lambda$ in the (new or known) currents appearing
in the latter algebra ii). Then after we take the above $\lambda\rightarrow 0$
limit in this algebra ii), we still have the same former algebra i) at
$\lambda=0$ because these “additional” terms vanish in this limit. This is
another deformed algebra. Therefore, it is nontrivial to determine the
“additional” current terms having the $\lambda$ factor explicitly for general
$h_{1}$ and $h_{2}$. They can appear as either the new currents with the
coefficients having the $\lambda$ factor or previously known currents with the
structure constants having the $\lambda$ factor explicitly.

Recently [9], by considering the particular number $K=2$ of free fields above,
the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$ algebra is
studied. For low weights, the explicit OPEs between the currents of ${\cal
N}=4$ multiplet are obtained. All the structure constants appearing on the
right hand sides of these OPEs are given in terms of the above deformation
parameter $\lambda$ explicitly but the general behavior of those on the
weights is not known.

In this paper, we continue to study the structure of the ${\cal N}=4$
supersymmetric linear $W_{\infty}[\lambda]$ algebra found in [9]. We try to
determine the ${\cal N}=4$ multiplet for any weight $h$ in terms of above free
field operators. The corresponding five components are determined explicitly
for arbitrary weight $h$. After the explicit form of the ${\cal N}=4$
multiplet is obtained, we perform their OPEs by using the defining OPEs
between the free fields. In doing this, we should use the previous results in
[8] for the OPEs between the nonsinglet currents, as an intermediate step. We
will observe the appearance of the extra structures (described before) on the
right hand sides of the (anti)commutators for the particular weights $h_{1}$
and $h_{2}$. Eventually we will determine the (anti)commutator relations
between the currents of the ${\cal N}=4$ multiplet, as in the abstract. In the
various footnotes, we emphasize that the extra structures on the right hand
sides of the (anti)commutators arise for the specific $h_{1}$ and $h_{2}$.

In section $2$, we construct the ${\cal N}=4$ multiplet in terms of free
fields for general weight $h$. In section $3$, the explicit (anti)commutator
relations between the nonsinglet currents for the weights $h_{1}$ and $h_{2}$
satisfying some constraints are obtained. In section $4$, the fundamental
commutator relations between the ${\cal N}=4$ multiplets of $SO(4)$ singlets
or nonsinglets are determined. In Appendices, the details appearing in
sections $3$ and $4$ are described. In particular, the remaining ten
(anti)commutators for the ${\cal N}=4$ multiplet are given here. We are using
the Thielemans package [10] with a mathematica [11].

We summarize what we have obtained as follows. At the vanishing deformation
parameter $\lambda=0$, the complete structure of the ${\cal N}=4$
supersymmetric linear $W_{\infty}[\lambda=0]$ algebra is given by (4.7),
(4.9), (4.11), (4.12) and (4.16) in addition to Appendices (G.4), (G.5),
(G.9), (G.11), (G.15), (G.18), (G.19), (G.23), (G.25) and (G.29) where all the
$\lambda$ dependence appearing on the right hand sides is gone by putting
$\lambda=0$. Compared to the previous work in [12], due to the presence of the
weights $1$ and $\frac{1}{2}$ currents mentioned before, in general, they do
appear on the right hand sides of above (anti)commutators at $\lambda=0$.
However, in the most of the examples, the structure constants appearing in
these currents are vanishing at $\lambda=0$. See also the footnotes in
sections $3$ and $4$. It is an open problem whether the present algebra at
$\lambda=0$ can be reduced to the one of [12] by decoupling the above weights
$1$ and $\frac{1}{2}$ currents. For nonzero $\lambda$, still the above
(anti)commutator relations between the currents can be used for the weights
$h_{1}$ and $h_{2}$ satisfying some constraints. From the analysis in the
footnotes of section $4$, this algebra at nonzero $\lambda$ is different from
the one in [13]. In other words, they have common algebra at $\lambda=0$
(their structure constants are the same) and for nonzero $\lambda$, one
deformed algebra is given by [13] and another one is the algebra obtained in
this paper.

Then what happens for the generic weights $h_{1}$ and $h_{2}$? First of all,
the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$ algebra is linear
in the sense that the right hand sides of the OPEs between the currents
contain all the possible current terms which are linear. Furthermore, the
${\cal N}=4$ multiplet is given by (2.15), (2.16), (2.17), (2.18) and (2.19).
Then we can perform any OPE inside the Thielemans package [10] using the
explicit forms of the ${\cal N}=4$ multiplet. Of course, from the beginning we
should fix the weights $h_{1}$ and $h_{2}$ within the package. Then the
possible poles are given by the highest singular term which is
$(h_{1}+h_{2})$-th order pole, the next singular term which is
$(h_{1}+h_{2}-1)$-th order pole, and so on until the first order pole. Then
the next step is to rewrite all the singular terms in terms of the components
of the ${\cal N}=4$ multiplet of $SO(4)$ nonsinglets or singlets. This is
straightforward because the possible current terms at the particular singular
term are known. The weight $h$ is fixed and the possible currents can be
determined. They consist of the current having that weight $h$, the current
having the weight $(h-1)$ with one derivative, $\cdots$, and the current
having the weight $1$ with $(h-1)$ derivative. See also Appendix $C$ for the
specific examples.

Now we introduce the arbitrary coefficients on these possible current terms
and solve the linear equations for these unknown coefficients by requiring
that the algebra is closed in the sense that the right hand side of the OPE
contains the components of ${\cal N}=4$ multiplet. It will turn out that they
can be determined explicitly in terms of the deformation parameter $\lambda$.
The point is, according to the result of this paper, that there exists a
critical singular term where the possible current of weight $h_{c}$ is
allowed. If $h\leq h_{c}$, then we expect to have the additional structures
(either the presence of new currents or the different $\lambda$ dependent
structure constants in the previously known currents) on the right hand side
of the OPE. In this case, although the structure constants are known in terms
of the $\lambda$ for fixed $h_{1}$ and $h_{2}$, their explicit expressions for
generic $h_{1}$ and $h_{2}$ are not known so far. On the other hand, if $h\geq
h_{c}$, then still we can use the previous (anti)commutator relations
described in the previous paragraph, for nonzero $\lambda$ without any
modifications. In this case, all the structure constants are known and they
are given by those in Appendix $A$.

Let us list some future directions along the line of the present paper.

$\bullet$ The ${\cal N}=4$ superspace OPE

Although we have found $15$ (anti)commutator relations explicitly, it is nice
to observe its ${\cal N}=4$ superspace description. In order to perform the
${\cal N}=4$ superspace approach, we need to rewrite the above fundamental
OPEs in (4.7), (4.9), (4.11), (4.12) and (4.16) such that the second element
with the coordinate $w$ on the left hand side of the OPE should be the lowest
component of the ${\cal N}=4$ multiplet. That is, they are given by
$\big{[}(\Phi^{(h_{1})}_{0})_{m},(\Phi^{(h_{2})}_{0})_{n}\big{]}$,
$\big{[}(\Phi^{(h_{1}),i}_{\frac{1}{2}})_{r},(\Phi^{(h_{2})}_{0})_{n}\big{]}$,
$\big{[}(\Phi^{(h_{1}),ij}_{1})_{m},(\Phi^{(h_{2})}_{0})_{n}\big{]}$,
$\big{[}(\Phi^{(h_{1}),i}_{\frac{3}{2}})_{r},(\Phi^{(h_{2})}_{0})_{n}\big{]}$,
and $\big{[}(\Phi^{(h_{1})}_{2})_{m},(\Phi^{(h_{2})}_{0})_{n}\big{]}$ in the
commutators. After these are obtained, then it is straightforward to express
them in the ${\cal N}=4$ superspace. For consistency check, it is obvious to
extract the remaining $10$ (anti)commutator relations (or its corresponding
OPEs) from the above ${\cal N}=4$ superspace description. In other words, we
do not have to calculate the remaining (anti)commutators separately and this
is the power of ${\cal N}=4$ supersymmetry.

$\bullet$ The complete structure constants for any $h_{1}$ and $h_{2}$ for
nonzero $\lambda$

One way to determine these is that it is better to consider the modes of the
currents in terms of those in the free fields. By simplifying the
(anti)commutator relations in terms of the modes of the free fields, we can
express them in terms of several (anti)commutator relations according to the
decomposition of the (anti)commutator in quantum mechanics. Then we can use
the corresponding (anti)commutator relations for (2.1). We can try to obtain
the general structure by fixing the weights $h_{1}$ and $h_{2}$. It is rather
nontrivial to determine the structure constants for generic $h_{1}$ and
$h_{2}$ by varying them.

$\bullet$ Realization of the present algebra in the celestial conformal field
theory

At the vanishing deformation parameter $\lambda=0$, the algebra is known
completely. In other words, the structure constants are given in Appendix $A$
by inserting the $\lambda=0$. We are left with another deformation parameter
$q$. As described before, we realize that the $w_{1+\infty}$ algebra can be
obtained from the $SO(4)$ singlet currents via proper contractions of the
currents with vanishing $q$ limit at $\lambda=0$. It would be interesting to
observe whether there exists any realization of the present algebra in the
celestial conformal field theory, along the line of [14, 15, 16, 17, 18, 19,
20], or not.

##  2  The ${\cal N}=4$ multiplet

The ${\cal N}=4$ multiplet for any weight $h$ is described by using the free
bosonic and fermionic fields.

###  2.1 Review

The $\beta\,\gamma$ and $b\,c$ systems satisfy the following operator product
expansions

$\displaystyle\gamma^{i,\bar{a}}(z)\,\beta^{\bar{j},b}(w)=\frac{1}{(z-w)}\,\delta^{i\bar{j}}\,\delta^{\bar{a}b}+\cdots\,,\qquad
c^{i,\bar{a}}(z)\,b^{\bar{j},b}(w)=\frac{1}{(z-w)}\,\delta^{i\bar{j}}\,\delta^{\bar{a}b}+\cdots\,.$
(2.1)

The fundamental indices $a,b$ of $SU(2)$ run over $a,b=1,2$ while the
antifundamental indices $\bar{a},\bar{b}$ of $SU(2)$ run over
$\bar{a},\bar{b}=1,2$. Similarly the fundamental indices $i,j$ of $SU(N)$ run
over $i,j=1,2,\cdots,N$ and the antifundamental indices $\bar{i},\bar{j}$ of
$SU(N)$ run over $\bar{i},\bar{j}=1,2,\cdots,N$. The $(\beta,\gamma)$ fields
are bosonic operators and the $(b,c)$ fields are fermionic operators.

Then the $SU(N)$ singlet currents (the generalization of [1, 2]) can be
obtained by taking the bilinears of above free fields with a summation over
the (anti)fundamental indices of $SU(N)$ as follows:

$\displaystyle V_{\lambda,\bar{a}b}^{(h)+}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{h-1}\,a^{i}(h,\lambda)\,\partial^{h-1-i}\,((\partial^{i}\,\beta^{\bar{l}b})\,\delta_{l\bar{l}}\,\gamma^{l\bar{a}})+\sum_{i=0}^{h-1}\,a^{i}(h,\lambda+\frac{1}{2})\,\partial^{h-1-i}\,((\partial^{i}\,b^{\bar{l}b})\,\delta_{l\bar{l}}\,c^{l\bar{a}})\,,$
$\displaystyle V_{\lambda,\bar{a}b}^{(h)-}$ $\displaystyle=$
$\displaystyle-\frac{(h-1+2\lambda)}{(2h-1)}\,\sum_{i=0}^{h-1}\,a^{i}(h,\lambda)\,\partial^{h-1-i}\,((\partial^{i}\,\beta^{\bar{l}b})\,\delta_{l\bar{l}}\,\gamma^{l\bar{a}})$
$\displaystyle+$
$\displaystyle\frac{(h-2\lambda)}{(2h-1)}\,\sum_{i=0}^{h-1}\,a^{i}(h,\lambda+\frac{1}{2})\,\partial^{h-1-i}\,((\partial^{i}\,b^{\bar{l}b})\,\delta_{l\bar{l}}\,c^{l\bar{a}})\,,$
$\displaystyle Q_{\lambda,\bar{a}b}^{(h)+}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{h-1}\,\alpha^{i}(h,\lambda)\,\partial^{h-1-i}\,((\partial^{i}\,\beta^{\bar{l}b})\,\delta_{l\bar{l}}\,c^{l\bar{a}})-\sum_{i=0}^{h-2}\,\beta^{i}(h,\lambda)\,\partial^{h-2-i}\,((\partial^{i}\,b^{\bar{l}b})\,\delta_{l\bar{l}}\,\gamma^{l\bar{a}})\,,$
$\displaystyle Q_{\lambda,\bar{a}b}^{(h)-}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{h-1}\,\alpha^{i}(h,\lambda)\,\partial^{h-1-i}\,((\partial^{i}\,\beta^{\bar{l}b})\,\delta_{l\bar{l}}\,c^{l\bar{a}})+\sum_{i=0}^{h-2}\,\beta^{i}(h,\lambda)\,\partial^{h-2-i}\,((\partial^{i}\,b^{\bar{l}b})\,\delta_{l\bar{l}}\,\gamma^{l\bar{a}})\,.$
(2.2)

Note that the weights of these currents are given by $h$, $h$,
$(h-\frac{1}{2})$ and $(h-\frac{1}{2})$ respectively. The conformal weights of
$(\beta,\gamma)$ fields are given by $(\lambda,1-\lambda)$ while conformal
weights of $(b,c)$ fields are given by
$(\frac{1}{2}+\lambda,\frac{1}{2}-\lambda)$. The above weights for the
currents do not depend on the deformation parameter $\lambda$ due to the
particular combinations of the free fields. By counting the number of
(anti)fundamental indices, there exist four components labeled by
$(\bar{a}b)=(11,12,21,22)$ in each current.

The relative coefficients appearing in (2.2) depend on the conformal weight
$h$ and deformation parameter $\lambda$ explicitly and they are given by the
binomial coefficients and the rising Pochhammer symbols where $(a)_{n}\equiv
a(a+1)\cdots(a+n-1)$ as follows [1, 2]:

$\displaystyle a^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-1\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-1-i}}{(h+i)_{h-1-i}}\,,\qquad
0\leq i\leq(h-1)\,,$ (2.5)
$\displaystyle\alpha^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-1\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-1-i}}{(h+i-1)_{h-1-i}}\,,\qquad
0\leq i\leq(h-1)\,,$ (2.8)
$\displaystyle\beta^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-2\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-2-i}}{(h+i)_{h-2-i}}\,,\qquad
0\leq i\leq(h-2)\,.$ (2.11)

Let us consider the following currents consisting of $(b,c)$ fields,
$(\beta,\gamma)$ fields, $(\gamma,b)$ fields and $(\beta,c)$ fields
respectively by taking the linear combinations of (2.2) with the help of
(2.11)

$\displaystyle W^{\lambda,\bar{a}b}_{F,h}(b,c)$ $\displaystyle=$
$\displaystyle\frac{2^{h-3}(h-1)!}{(2h-3)!!}\,\,\frac{(-1)^{h}}{\sum_{i=0}^{h-1}\,a^{i}(h,\frac{1}{2})}\,\Bigg{[}\frac{(h-1+2\lambda)}{(2h-1)}\,V_{\lambda,\bar{a}b}^{(h)+}+V_{\lambda,\bar{a}b}^{(h)-}\Bigg{]}\,,$
$\displaystyle W^{\lambda,\bar{a}b}_{B,h}(\beta,\gamma)$ $\displaystyle=$
$\displaystyle\frac{2^{h-3}\,h!}{(2h-3)!!}\,\frac{(-1)^{h}}{\sum_{i=0}^{h-1}\,a^{i}(h,0)}\,\Bigg{[}\frac{(h-2\lambda)}{(2h-1)}\,V_{\lambda,\bar{a}b}^{(h)+}-V_{\lambda,\bar{a}b}^{(h)-}\Bigg{]}\,,$
$\displaystyle Q^{\lambda,\bar{a}b}_{h+\frac{1}{2}}(\gamma,b)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\frac{2^{h-\frac{1}{2}}h!}{(2h-1)!!}\,\frac{(-1)^{h+1}\,h}{\sum_{i=0}^{h-1}\,\beta^{i}(h+1,0)}\,\Bigg{[}Q_{\lambda,\bar{a}b}^{(h+1)-}-Q_{\lambda,\bar{a}b}^{(h+1)+}\Bigg{]}\,,$
$\displaystyle\bar{Q}^{\lambda,b\bar{a}}_{h+\frac{1}{2}}(\beta,c)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\frac{2^{h-\frac{1}{2}}h!}{(2h-1)!!}\,\frac{(-1)^{h+1}}{\sum_{i=0}^{h}\,\alpha^{i}(h+1,0)}\,\Bigg{[}Q_{\lambda,\bar{a}b}^{(h+1)-}+Q_{\lambda,\bar{a}b}^{(h+1)+}\Bigg{]}\,.$
(2.12)

The overall coefficients do not depend on the deformation parameter $\lambda$.
Then we have eight bosonic currents for the weight $h=1,2,\cdots$ and eight
fermionic currents for the weight
$h+\frac{1}{2}=\frac{3}{2},\frac{5}{2},\cdots$ as well as four fermionic
currents $\bar{Q}^{\lambda,b\bar{a}}_{\frac{1}{2}}$ of the weight
$\frac{1}{2}$ in (2.12). Note that four fermionic currents
$Q^{\lambda,\bar{a}b}_{\frac{1}{2}}$ of the weight $\frac{1}{2}$ are
identically zero.

The stress energy tensor of weight $2$ is given by

$\displaystyle L$ $\displaystyle=$
$\displaystyle\Big{(}W^{\lambda,\bar{a}a}_{\mathrm{B},2}+W^{\lambda,\bar{a}a}_{\mathrm{F},2}\Big{)}\,,$
(2.13)

which can be written as $V_{\lambda,\bar{a}a}^{(2)+}$. The corresponding
central charge is

$\displaystyle c_{cen}=6\,N\,(1-4\lambda)\,,$ (2.14)

which depends on the deformation parameter $\lambda$ explicitly. The above
bosonic and fermionic currents in (2.12) are quasiprimary operators under the
stress energy tensor (2.13) by using the defining OPEs in (2.1). The central
charge (2.14) becomes $c_{cen}=6N$ at $\lambda=0$.

###  2.2 The ${\cal N}=4$ multiplet

####  2.2.1 The lowest component

It is known, in [9], that the lowest components $\Phi_{0}^{(h)}$ for the
weights $h=1,2,3$ and $4$ have their explicit $\lambda$ dependences
$(h-2\lambda)$ and $(h-1+2\lambda)$ in their relative coefficients. Then the
question is how we determine these relative coefficients for arbitrary weight
$h$. We realize that there exists an additional overall factor $-4$ from the
weight $h$ to the weight $(h+1)$ in (2.12). Moreover, the denominator of the
overall factor can be extracted easily and is given by $\frac{1}{(2h+1)}$ in
terms of the weight $h$. We expect to have the factor $(-4)^{h}$ from the
above analysis. The other numerical ($h$ independent) factor can appear in
general. This can be fixed only after we calculate the OPE between this lowest
component and itself and obtain the central term. We will compute this central
term later. For the time being we simply write down the following form

$\displaystyle\Phi_{0}^{(h)}$ $\displaystyle=$
$\displaystyle\frac{(-4)^{h-2}}{(2h-1)}\,\Bigg{[}-(h-2\lambda)\,W^{\lambda,\bar{a}a}_{\mathrm{F},h}+(h-1+2\lambda)\,W^{\lambda,\bar{a}a}_{\mathrm{B},h}\Bigg{]}\,.$
(2.15)

For the weights $h=1,2,3,4$, we can observe that the corresponding numerical
values appearing in the relative coefficients of (2.15) can be seen from the
ones in [9]. The normalization in (2.15) is different from the one in [9]
where there appear the additional numerical factors $16,8,12,24$ for the
weights $h=1,2,3,4$ respectively. At the moment, it is not easy to figure out
the exact $h$ dependence from these values. In other words, we take the $h$
dependence as in (2.15) together with the additional numerical factor
$(-4)^{-2}$. As explained before, the overall factor in (2.15) can be
determined by the normalization of the highest order singular term in the OPE
between the $\Phi_{0}^{(h)}$ and itself.

####  2.2.2 The second components

From the observation of [9] for the weights $h=1,2,3,4$, the second components
contain the various fermionic currents and their relative coefficients are
common for any weights $h=1,2,3,4$. This implies that it is natural to take
these relative coefficients for any $h$ and the question is how we obtain the
overall numerical factor. By taking the normalization of (2.15) we can extract
the overall factors for the weights $h=1,2,3,4$ and they are given by
$-\frac{1}{16}\times 1=-\frac{1}{16}$, $-\frac{1}{8}\times(-2)=\frac{1}{4}$,
$-\frac{1}{12}\times 12=-1$ and $-\frac{1}{24}\times 96=4$ respectively. We
can easily see that there exists $(-4)^{h}$ dependence when we increase the
weight by $1$. Therefore, the general expression for the weight $h$ is given
by $4(-4)^{h-4}$ which covers the above numerical values for the weights
$h=1,2,3,4$. Then we can write down the corresponding second components as
follows:

$\displaystyle\Phi^{(h),1}_{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{1}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{h+\frac{1}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{h+\frac{1}{2}}-2\,Q^{\lambda,22}_{h+\frac{1}{2}}$
$\displaystyle+$ $\displaystyle
2\,\bar{Q}^{\lambda,11}_{h+\frac{1}{2}}+2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{h+\frac{1}{2}}+i\sqrt{2}\,\bar{Q}^{\lambda,21}_{h+\frac{1}{2}}-\bar{Q}^{\lambda,22}_{h+\frac{1}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),2}_{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{1}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{h+\frac{1}{2}}-2\,Q^{\lambda,22}_{\frac{3}{2}}+2\,\bar{Q}^{\lambda,11}_{h+\frac{1}{2}}+2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{h+\frac{1}{2}}-\bar{Q}^{\lambda,22}_{h+\frac{1}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),3}_{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{1}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{h+\frac{1}{2}}-2\,Q^{\lambda,22}_{h+\frac{1}{2}}+2\,\bar{Q}^{\lambda,11}_{h+\frac{1}{2}}+i\sqrt{2}\,\bar{Q}^{\lambda,21}_{h+\frac{1}{2}}-\bar{Q}^{\lambda,22}_{h+\frac{1}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),4}_{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-\frac{1}{2}\,Q^{\lambda,11}_{h+\frac{1}{2}}-Q^{\lambda,22}_{h+\frac{1}{2}}+\bar{Q}^{\lambda,11}_{h+\frac{1}{2}}+\frac{1}{2}\,\bar{Q}^{\lambda,22}_{h+\frac{1}{2}}\Bigg{]}\,.$
(2.16)

Or we can calculate the OPEs between the supersymmetry generators and the
lowest component and read off the first order pole which will provide the
second components in (2.16). We will calculate the central terms coming from
the highest order pole between the second components and itself later. As
described before, once we fix this central term, then the overall factor in
(2.16) where we use the normalization in (2.15) can be determined.

####  2.2.3 The third components

The third components contain the various bosonic currents and their relative
coefficients are equal to each other for any weights $h=1,2,3,4$ in the
analysis of [9]. Then we obtain the following results by replacing them with
the corresponding expressions for arbitrary weight $h$ with the above overall
factors in previous section

$\displaystyle\Phi^{(h),12}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}2i\,W^{\lambda,11}_{\mathrm{B},h+1}-\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},h+1}-2i\,\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle+$ $\displaystyle
2i\,W^{\lambda,11}_{\mathrm{F},h+1}-2\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},h+1}-2i\,W^{\lambda,22}_{\mathrm{F},h+1}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),13}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},h+1}+4\sqrt{2}\,W^{\lambda,21}_{\mathrm{B},h+1}+2i\,\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle-$ $\displaystyle
2i\,W^{\lambda,11}_{\mathrm{F},h+1}+2\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},h+1}+2i\,W^{\lambda,22}_{\mathrm{F}+h+1}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),14}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}2\,W^{\lambda,11}_{\mathrm{B},h+1}+i\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},h+1}+4i\sqrt{2}\,\,W^{\lambda,21}_{\mathrm{B},h+1}-2\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle-$ $\displaystyle
2\,W^{\lambda,11}_{\mathrm{F},h+1}-2i\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},h+1}-2i\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},h+1}+2\,W^{\lambda,22}_{\mathrm{F},h+1}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),23}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-2\,W^{\lambda,11}_{\mathrm{B},h+1}-i\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},h+1}-4i\sqrt{2}\,\,W^{\lambda,21}_{\mathrm{B},h+1}+2\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle-$ $\displaystyle
2\,W^{\lambda,11}_{\mathrm{F},h+1}-2i\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},h+1}-2i\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},h+1}+2\,W^{\lambda,22}_{\mathrm{F},h+1}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),24}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},h+1}+4\sqrt{2}\,W^{\lambda,21}_{\mathrm{B},h+1}+2i\,\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle+$ $\displaystyle
2i\,W^{\lambda,11}_{\mathrm{F},h+1}-2\sqrt{2}\,W^{\lambda,21}_{\mathrm{F},h+1}-2i\,W^{\lambda,22}_{\mathrm{F},h+1}\,\Bigg{]}\,,$
$\displaystyle\Phi^{(h),34}_{1}$ $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-2i\,W^{\lambda,11}_{\mathrm{B},h+1}+\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},h+1}+2i\,\,W^{\lambda,22}_{\mathrm{B},h+1}$
(2.17) $\displaystyle+$ $\displaystyle
2i\,W^{\lambda,11}_{\mathrm{F},h+1}-2\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},h+1}-2i\,W^{\lambda,22}_{\mathrm{F},h+1}\,\Bigg{]}\,.$

In principle, the OPEs between the supersymmetry generators of ${\cal N}=4$
superconformal algebra and the second components and the first order pole will
provide the third components in (2.17). The central terms coming from the
highest order pole between the third components and itself will be determined
later. Note that we can express the linear combination of
$W_{B,h+1}^{\lambda,\bar{a}b}$ and another linear combination of
$W_{F,h+1}^{\lambda,\bar{a}b}$ in terms of $\Phi_{1}^{(h),ij}$ and
$\frac{1}{2}\,\varepsilon^{ijkl}\,\Phi_{1}^{(h),kl}$. For example, the
$\Phi_{1}^{(h),12}$ and the $\Phi_{1}^{(h),34}$ look similar to each other in
the sense that the field contents are the same and half of them have opposite
signs. By adding or subtracting these two relations, the two independent field
contents can be written in terms of the third components of ${\cal N}=4$
multiplet as above.

####  2.2.4 The fourth components

The fourth components contain the various fermionic currents and their
relative coefficients are equal to each other for any weights $h=1,2,3,4$ in
the analysis of [9]. By replacing them with the corresponding expressions for
arbitrary weight $h$ with the above overall factors in previous section we
determine the following results as follows:

$\displaystyle\tilde{\Phi}^{(h),1}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(h),1}_{\frac{3}{2}}-\frac{1}{(2h+1)}\,(1-4\lambda)\,\partial\,\Phi^{(h),1}_{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}-\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{3}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{h+\frac{3}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{h+\frac{3}{2}}-2\,Q^{\lambda,22}_{h+\frac{3}{2}}$
$\displaystyle-$ $\displaystyle
2\,\bar{Q}^{\lambda,11}_{h+\frac{3}{2}}-2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{h+\frac{3}{2}}-i\sqrt{2}\,\bar{Q}^{\lambda,21}_{h+\frac{3}{2}}+\bar{Q}^{\lambda,22}_{h+\frac{3}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(h),2}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(h),2}_{\frac{3}{2}}-\frac{1}{(2h+1)}\,(1-4\lambda)\,\partial\,\Phi^{(h),2}_{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}\frac{i}{2}\,\Big{(}\,Q^{\lambda,11}_{h+\frac{3}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{h+\frac{3}{2}}-2\,Q^{\lambda,22}_{h+\frac{3}{2}}-2\,\bar{Q}^{\lambda,11}_{h+\frac{3}{2}}-2i\sqrt{2}\,\bar{Q}^{\lambda,12}_{h+\frac{3}{2}}+\bar{Q}^{\lambda,22}_{h+\frac{3}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(h),3}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(h),3}_{\frac{3}{2}}-\frac{1}{(2h+1)}\,(1-4\lambda)\,\partial\,\Phi^{(h),3}_{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}\frac{i}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{3}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{h+\frac{3}{2}}-2\,Q^{\lambda,22}_{h+\frac{3}{2}}-2\,\bar{Q}^{\lambda,11}_{h+\frac{3}{2}}-i\sqrt{2}\,\bar{Q}^{\lambda,21}_{h+\frac{3}{2}}+\bar{Q}^{\lambda,22}_{h+\frac{3}{2}}\Big{)}\,\Bigg{]}\,,$
$\displaystyle\tilde{\Phi}^{(h),4}_{\frac{3}{2}}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(h),4}_{\frac{3}{2}}-\frac{1}{(2h+1)}\,(1-4\lambda)\,\partial\,\Phi^{(h),4}_{\frac{1}{2}}$
(2.18) $\displaystyle=$ $\displaystyle
4\,(-4)^{h-4}\,\Bigg{[}\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{3}{2}}+2\,Q^{\lambda,22}_{h+\frac{3}{2}}+2\,\bar{Q}^{\lambda,11}_{h+\frac{3}{2}}+\bar{Q}^{\lambda,22}_{h+\frac{3}{2}}\Big{)}\,\Bigg{]}\,.$

Compared to the $\Phi_{\frac{3}{2}}^{(h),i}$ which belongs to the components
of the ${\cal N}=4$ multiplet, the $\tilde{\Phi}_{\frac{3}{2}}^{(h),i}$ in
(2.18) are quasiprimary fields under the stress energy tensor (2.13). The
$\Phi_{\frac{3}{2}}^{(h),i}$ and the $\Phi_{\frac{1}{2}}^{(h+1),i}$ by
considering that the weight $h$ in (2.16) is replaced with the weight $(h+1)$
look similar to each other in the sense that the field contents are the same
and half of them have opposite signs. By adding or subtracting these two
relations as before, the two independent field contents can be written in
terms of the second components of the $(h+1)$-th ${\cal N}=4$ multiplet and
the fourth components of the $h$-th ${\cal N}=4$ multiplet. We expect that the
OPEs between the supersymmetry generators of ${\cal N}=4$ superconformal
algebra and the third components will provide the fourth components in (2.18).

####  2.2.5 The last component

Finally we describe the last component for arbitrary weight $h$ as follows:

$\displaystyle\tilde{\Phi}^{(h)}_{2}$ $\displaystyle\equiv$
$\displaystyle\Phi^{(h)}_{2}-\frac{1}{(2h+1)}\,(1-4\lambda)\,\partial^{2}\,\Phi^{(h)}_{0}=4\,(-4)^{h-4}\,\Bigg{[}-2\,\Big{(}W^{\lambda,\bar{a}a}_{\mathrm{B},h+2}+W^{\lambda,\bar{a}a}_{\mathrm{F},h+2}\Big{)}\Bigg{]}.$
(2.19)

Under the stress energy tensor (2.13), this is a quasiprimary operator. The
OPEs between the supersymmetry generators of ${\cal N}=4$ superconformal
algebra and the fourth components will provide the last component in (2.19).
By replacing $h$ with $(h+2)$ in (2.15), we can express
$W^{\lambda,\bar{a}a}_{\mathrm{B},h+2}$ and
$W^{\lambda,\bar{a}a}_{\mathrm{F},h+2}$ in terms of $\Phi_{0}^{(h+2)}$ and
$\tilde{\Phi}^{(h)}_{2}$ by simple linear combinations as before.

Therefore, the ${\cal N}=4$ multiplet is summarized by (2.15), (2.16), (2.17),
(2.18) and (2.19) together with (2.2), (2.11) and (2.12). Their algebra will
be obtained explicitly by using the defining relations in (2.1).

##  3  The ${\cal N}=4$ supersymmetric linear $W_{\infty}^{2,2}$ algebra
between the adjoints and the bifundamentals under the $U(2)\times U(2)$
symmetry

In order to obtain the algebra between (2.15), (2.16), (2.17), (2.18) and
(2.19), it is necessary to determine the algebra between the currents in
(2.12). In the footnotes, we present some examples where there are extra
structures (described in the introduction) on the right hand sides of the
(anti)commutator relations for the specific weights $h_{1}$ and $h_{2}$.

###  3.1  The (anti)commutator relations between the nonsinglet currents

Let us consider the algebra between the currents consisting of $(b,c)$ fields
in (2.12). By multiplying the Pauli matrix of $SU(2)$ with the additional
factor $\frac{1}{2}$ properly and summing over the indices $\bar{a}$ and $b$
as in [9], we can construct the three fundamentals of $SU(2)$. By multiplying
the Kronecker delta (or $2\times 2$ identity matrix) with the contractions of
the indices, we obtain the singlet of $SU(2)$. First of all, in $SU(2)$, there
is no symmetric $d^{\hat{A}\hat{B}\hat{C}}$ symbols.

####  3.1.1 The commutator relation with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero
$\lambda$

Then we can associate
$(W^{\lambda,12}_{\mathrm{F},h}+W^{\lambda,21}_{\mathrm{F},h})$,
$i\,(W^{\lambda,12}_{\mathrm{F},h}-W^{\lambda,21}_{\mathrm{F},h})$ and
$(W^{\lambda,11}_{\mathrm{F},h}-W^{\lambda,22}_{\mathrm{F},h})$ with the
triplets $W^{\lambda,\hat{A}=1}_{\mathrm{F},h}$,
$W^{\lambda,\hat{A}=2}_{\mathrm{F},h}$, and
$W^{\lambda,\hat{A}=3}_{\mathrm{F},h}$ of $SU(2)$ respectively. Moreover, the
$(W^{\lambda,11}_{\mathrm{F},h}+W^{\lambda,22}_{\mathrm{F},h})=W^{\lambda,\bar{a}a}_{\mathrm{F},h}$
plays the role of the singlet $W^{\lambda,\hat{A}=0}_{\mathrm{F},h}\equiv
W^{\lambda}_{\mathrm{F},h}$ of $SU(2)$.

One of the commutator relations in [9] can be written as the following
commutator relation 222 We can calculate the OPE between
$W^{\lambda,\hat{A}=1}_{\mathrm{F},h_{1}=6}(z)$ and
$W^{\lambda,\hat{B}=1}_{\mathrm{F},h_{2}=4}(w)$ where $h_{1}=h_{2}+2$ and read
off the ninth order pole which has the structure constant
$\frac{131072}{5}(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)$
appearing in the current $W^{\lambda}_{\mathrm{F},h_{1}+h_{2}-2-h=1}(w)$. In
this case, the weight $h$ is given by odd number $h=7$. We do not see this
term from the commutator (3.4) because the even $h$ appears in the last term
of (3.4). This implies that if the weights $h_{1}$ and $h_{2}$ do not satisfy
the above constraint ($h_{1}=h_{2}$ or $h_{1}=h_{2}\pm 1$) for nonzero
$\lambda$, then we cannot use the formula in (3.4) fully and there appear the
extra terms on the right hand sides of the corresponding OPE. Because the
above extra factor contains $\lambda$, there will be no problem for vanishing
$\lambda$ when we use (3.4). It would be interesting to obtain the above
commutator for generic $h_{1}$ and $h_{2}$. It seems that there is a critical
singular term in the sense that we still have the structure of (3.4) for the
poles less than this critical singular term. Of course, for the poles greater
than the critical singular term there exist other extra terms in general.

$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,\hat{B}}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,i\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
(3.3) $\displaystyle+\\!$ $\displaystyle\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{F}(h_{1},h_{2},\lambda)\,\delta^{\hat{A}\hat{B}}\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,\,\delta^{\hat{A}\hat{B}}\,(W^{\lambda}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,.$
(3.4)

The structure constant is given by (A.9).

Let us calculate the central term in (3.4) explicitly. One of the reasons why
we are doing this is that we have not seen any literatures which provides all
the details in the calculation and it is useful to observe the general
structure of the computation of any OPEs including the free fields. In order
to calculate the highest singular term in the OPE
$V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$, we
need to calculate the central term in the OPE between
$V_{\lambda,\bar{a}b}^{(h_{1})+}(z)$ and
$\delta_{l\bar{l}}\,\gamma^{l\bar{c}}\,\partial^{i}\,\beta^{\bar{l}d}(w)$
coming from $V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$. It is known that the
following OPE satisfies

$\displaystyle V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,\gamma^{j\bar{c}}(x)$
$\displaystyle=$
$\displaystyle\delta_{b\bar{c}}\,\sum_{j=0}^{h_{1}-1}\,a^{j}(h_{1},\lambda)\,(-1)^{h_{1}}\,j!\,\sum_{t=0}^{j+1}\,(j+1-t)_{h_{1}-1-j}\,\frac{1}{t!}\,\frac{1}{(z-x)^{h_{1}-t}}\,\partial^{t}\,\gamma^{l\bar{a}}(x)$
(3.5) $\displaystyle+$ $\displaystyle\cdots.$

The next step is to calculate the OPE between
$\partial^{t}_{x}\,\gamma^{l\bar{a}}(x)$ appearing in the last factor in (3.5)
and $\partial^{i}_{w}\,\beta^{\bar{l}d}(w)$. We have the defining OPE relation
in (2.1). The multiple derivative with respect to $x$ acting on
$\frac{1}{(x-w)}$ can be obtained explicitly and the similar multiple
derivative with respect to $w$ acting on $\frac{1}{(x-w)}$ can be rewritten as
the corresponding multiple derivative with respect to $x$ acting on
$\frac{1}{(x-w)}$ with the number of minus signs. Then we obtain the following
result

$\displaystyle\partial^{t}_{x}\,\gamma^{l\bar{a}}(x)\,\partial^{i}_{w}\,\beta^{\bar{l}d}(w)=\frac{1}{(x-w)^{t+i+1}}\,(-1)^{t}\,(t+i)!\,\delta^{l\bar{l}}\,\delta^{d\bar{a}}+\cdots\,.$
(3.6)

Now we expand $\frac{1}{(z-x)^{h_{1}-t}}$ appearing in the second factor from
the last in (3.5) around $x=w$ by using the Taylor expansion. Then we obtain
that the coefficient of $(x-w)^{t+i+1}$ in the $\frac{1}{(z-x)^{h_{1}+i+1}}$
evaluated at $x=w$ is given by $\frac{1}{(t+i+1)!}\,(h_{1}-t)_{t+i+1}$. We are
left, by collecting the contributions from (3.5) and (3.6), with

$\displaystyle\Bigg{[}\delta_{b\bar{c}}\,\sum_{j=0}^{h_{1}-1}\,a^{j}(h_{1},\lambda)\,(-1)^{h_{1}}\,j!\,\sum_{t=0}^{j+1}\,(j+1-t)_{h_{1}-1-j}\,\frac{1}{t!}\Bigg{]}\,\Bigg{[}(-1)^{t}\,(t+i)!\,\delta^{l\bar{l}}\,\delta^{d\bar{a}}\Bigg{]}$
$\displaystyle\times\Bigg{[}\frac{1}{(t+i+1)!}\,(h_{1}-t)_{t+i+1}\Bigg{]}\,,$
(3.7)

in the $\frac{1}{(z-w)^{h_{1}+i+1}}$ term. After acting the derivative
$\partial_{w}^{h_{2}-1-i}$ from the remaining factor in the first part of
$V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$ on the $\frac{1}{(z-w)^{h_{1}+i+1}}$
term, we obtain $(h_{1}+i+1)_{h_{2}-1-i}$. By combining with (3.7), the final
contribution from the central terms in the OPE between
$V_{\lambda,\bar{a}b}^{(h_{1})+}(z)$ and the first part of
$V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$ can be written as

$\displaystyle
N\,\delta_{b\bar{c}}\,\delta_{d\bar{a}}\,\sum_{j=0}^{h_{1}-1}\,\sum_{i=0}^{h_{2}-1}\,\sum_{t=0}^{j+1}\,a^{j}(h_{1},\lambda)\,a^{i}(h_{2},\lambda)$
$\displaystyle\times\frac{j!\,(t+i)!}{t!\,(t+i+1)!}\,(-1)^{h_{1}+t}\,(j+1-t)_{h_{1}-1-j}\,(h_{1}-t)_{t+1+i}\,(h_{1}+1+i)_{h_{2}-1-i}\,.$
(3.8)

We can do the similar calculation for the contribution from the second part of
$V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$. In this case, we should use the
following intermediate result in [9]

$\displaystyle V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,c^{j\bar{c}}(x)$
$\displaystyle=$
$\displaystyle\delta_{b\bar{c}}\sum_{j=0}^{h_{1}-1}a^{j}(h_{1},\lambda+\frac{1}{2})(-1)^{h_{1}}j!\sum_{t=0}^{j+1}(j+1-t)_{h_{1}-1-j}\frac{1}{t!}\frac{1}{(z-x)^{h_{1}-t}}\partial^{t}c^{l\bar{c}}(w)$
(3.9) $\displaystyle+$ $\displaystyle\cdots.$

By starting with (3.9) and following the procedures in (3.6), (3.7) and (3.8)
we have described above, we obtain the following contribution

$\displaystyle-N\,\delta_{b\bar{c}}\,\delta_{d\bar{a}}\,\sum_{j=0}^{h_{1}-1}\,\sum_{i=0}^{h_{2}-1}\,\sum_{t=0}^{j+1}\,a^{j}(h_{1},\lambda+\frac{1}{2})\,a^{i}(h_{2},\lambda+\frac{1}{2})$
$\displaystyle\times\frac{j!\,(t+i)!}{t!\,(t+i+1)!}\,(-1)^{h_{1}+t}\,(j+1-t)_{h_{1}-1-j}\,(h_{1}-t)_{t+1+i}\,(h_{1}+1+i)_{h_{2}-1-i}\,.$
(3.10)

Therefore, we obtain the final central term, by adding (3.8) and (3.10), as
follows:

$\displaystyle
V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)\Bigg{|}_{\frac{1}{(z-w)^{h_{1}+h_{2}}}}=N\,\delta_{b\bar{c}}\,\delta_{d\bar{a}}\,\sum_{j=0}^{h_{1}-1}\,\sum_{i=0}^{h_{2}-1}\,\sum_{t=0}^{j+1}\,$
$\displaystyle\times\Bigg{(}a^{j}(h_{1},\lambda)\,a^{i}(h_{2},\lambda)-a^{j}(h_{1},\lambda+\frac{1}{2})\,a^{i}(h_{2},\lambda+\frac{1}{2})\Bigg{)}$
$\displaystyle\times\frac{j!\,(t+i)!}{t!\,(t+i+1)!}\,(-1)^{h_{1}+t}\,(j+1-t)_{h_{1}-1-j}\,(h_{1}-t)_{t+1+i}\,(h_{1}+1+i)_{h_{2}-1-i}\,.$
(3.11)

Due to the behavior of two Kronecker deltas, the central term is nonzero only
for the case where the second index of the first operator should equal to the
first index of the second operator and the first index of the first operator
should equal to the second index of the second operator on the left hand side.
In Appendix $B$, we present other central terms.

By realizing that the overall factor appearing in the first current of (2.12)
is given by $(-4)^{h-2}$, we can write down the central terms

$\displaystyle c_{F}(h_{1},h_{2},\lambda)$ $\displaystyle\equiv$
$\displaystyle(-4)^{h_{1}+h_{2}-4}\,\delta_{b\bar{a}}\,\delta_{d\bar{c}}\,\Bigg{[}\frac{(h_{1}-1+2\lambda)}{(2h_{1}-1)}\,\frac{(h_{2}-1+2\lambda)}{(2h_{2}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$
(3.12) $\displaystyle+$ $\displaystyle
V_{\lambda,\bar{a}b}^{(h_{1})-}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)+\frac{(h_{1}-1+2\lambda)}{(2h_{1}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)$
$\displaystyle+$
$\displaystyle\frac{(h_{2}-1+2\lambda)}{(2h_{2}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})-}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)\Bigg{]}_{\frac{1}{(z-w)^{h_{1}+h_{2}}}}\,,$

where the relation (3.11) and the relations in Appendix (B.1) are used. In the
last term of (3.12), we can use the central term of
$V_{\lambda,\bar{c}d}^{(h_{2})+}(z)\,V_{\lambda,\bar{a}b}^{(h_{1})-}(w)$ with
the extra factor $(-1)^{h_{1}+h_{2}}$.

In Appendices (C.1) and (C.2), we present the corresponding OPEs for
$h_{1}=h_{2}=4$ where the indices $\hat{A}$ and $\hat{B}$ are equal to each
other for the former while they are different from each other for the latter.
Compared to the commutator relation in (3.4), there exists $(-1)^{h-1}$
factor. Due to the Kronecker delta, the former corresponds to the last two
terms in (3.4) while the latter corresponds to the first term in (3.4). The
other four cases between the nonsinglet currents are checked explicitly and we
do not present them in this paper. The commutator relations between the
nonsinglet currents are given in [8]. The corresponding commutator relations
between the nonsinglet currents and the singlet currents can be determined
similarly.

####  3.1.2 The second commutator relation with $h_{1}=h_{2},h_{2}\pm 1$ for
nonzero $\lambda$

Similarly, we obtain the following commutator relation for the currents
consisting of $(\beta,\gamma)$ fields 333 As in the footnote 2, the OPE
between $W^{\lambda,\hat{A}=1}_{\mathrm{B},h_{1}=7}(z)$ and
$W^{\lambda,\hat{B}=1}_{\mathrm{B},h_{2}=3}(w)$ where $h_{1}=h_{2}+4$ can be
calculated and the ninth order pole contains the structure constant
$-\frac{524288}{11}(\lambda-1)\lambda(2\lambda-3)(2\lambda-1)(2\lambda+1)(\lambda^{2}-\lambda+5)$
appearing in the current $W^{\lambda}_{\mathrm{B},h_{1}+h_{2}-2-h=1}(w)$ and
the seventh order pole contains the structure constant
$-6144(\lambda-1)\lambda(2\lambda-3)(2\lambda-1)(2\lambda+1)$ appearing in the
current $W^{\lambda}_{\mathrm{B},h_{1}+h_{2}-2-h=3}(w)$. In this case, the
weight $h$ is given by odd number $h=7$ or $h=5$ while the corresponding dummy
variable $h$ is given by even number in (3.16). If the weights $h_{1}$ and
$h_{2}$ do not satisfy the above constraint ($h_{1}=h_{2}$ or $h_{1}=h_{2}\pm
1$) for nonzero $\lambda$, then we cannot use the formula in (3.16) exactly
because the extra terms on the right hand sides of the corresponding OPE
occur, compared to the $\lambda=0$ case. We observe that for the poles less
than the critical singular term mentioned in the footnote 2, the above
commutator can be used precisely and for the poles greater than that singular
term there exist extra terms on the right hand side of the OPE. Although these
extra terms can be obtained for fixed $h_{1}$ and $h_{2}$, at the moment,
those for general $h_{1}$ and $h_{2}$ are not known. It seems that as the
difference between $h_{1}$ and $h_{2}$ increases, the pole corresponding to
the critical singular term decreases.

$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{B},h_{1}})_{m},(W^{\lambda,\hat{B}}_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}}\,q^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n,\lambda)i\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}$
(3.15) $\displaystyle+\\!$ $\displaystyle\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{B}(h_{1},h_{2},\lambda)\,\delta^{\hat{A}\hat{B}}\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n,\lambda)\,\delta^{\hat{A}\hat{B}}\,(W^{\lambda}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}\,.$
(3.16)

The central term appearing in (3.16), by recalling the definition of the
second relation of (2.12), can be described by

$\displaystyle c_{B}(h_{1},h_{2},\lambda)$ $\displaystyle\equiv$
$\displaystyle(-4)^{h_{1}+h_{2}-4}\,\delta_{b\bar{a}}\,\delta_{d\bar{c}}\,\Bigg{[}\frac{(h_{1}-2\lambda)}{(2h_{1}-1)}\,\frac{(h_{2}-2\lambda)}{(2h_{2}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)$
(3.17) $\displaystyle+$ $\displaystyle
V_{\lambda,\bar{a}b}^{(h_{1})-}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)-\frac{(h_{1}-2\lambda)}{(2h_{1}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)$
$\displaystyle-$
$\displaystyle\frac{(h_{2}-2\lambda)}{(2h_{2}-1)}\,V_{\lambda,\bar{a}b}^{(h_{1})-}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})+}(w)\Bigg{]}_{\frac{1}{(z-w)^{h_{1}+h_{2}}}}\,.$

The previous relation (3.11) and the previous relations in Appendix (B.1) can
be used. As before, in the last term of (3.17), the central term of
$V_{\lambda,\bar{c}d}^{(h_{2})+}(z)\,V_{\lambda,\bar{a}b}^{(h_{1})-}(w)$ with
the extra factor $(-1)^{h_{1}+h_{2}}$ can be used. The relevant OPEs are given
in Appendices (C.3) and (C.4). The additional $(-1)^{h-1}$ factor appears in
the OPEs.

####  3.1.3 Other commutator relations with $h_{1}=h_{2},h_{2}+1$ for nonzero
$\lambda$

The remaining commutator relations between the bosonic currents and the
fermionic currents can be described as

$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{F},h_{1}})_{m},(Q^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,q^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\,\,\Bigg{(}i\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\lambda,\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\,\delta^{\hat{A}\hat{B}}\,(Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{)}\,,$
$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{B},h_{1}})_{m},(Q^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,q^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\,\Bigg{(}-i\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\lambda,\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\delta^{\hat{A}\hat{B}}\,(Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{)}\,,$
$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{F},h_{1}})_{m},(\bar{Q}^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-2}_{h=-1}\,q^{h}\,(-1)^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\,\Bigg{(}-i\,f^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\lambda,\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\,\delta^{\hat{A}\hat{B}}\,(\bar{Q}^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{)}\,,$
$\displaystyle\big{[}(W^{\lambda,\hat{A}}_{\mathrm{B},h_{1}})_{m},(\bar{Q}^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-2}_{h=-1}\,q^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\,\Bigg{(}i\,f^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\lambda,\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
(3.18) $\displaystyle+\\!$
$\displaystyle\,\delta^{\hat{A}\hat{B}}\,(\bar{Q}^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{)}\,.$

The corresponding OPEs for fixed $h_{1}$ and $h_{2}$ can be found in
Appendices (C.5), (C.6),(C.7), (C.8),(C.9), (C.10), (C.11) and (C.12). In the
last two commutator relations of (3.18), the upper limit of $h$ is given by
$h=h_{1}+h_{2}-2$ due to the presence of the lowest fermionic current
$\bar{Q}^{\lambda,b\bar{a}}_{\frac{1}{2}}$. The additional $(-1)^{h-1}$ factor
appears when we change the commutator relations into the corresponding OPEs
444 As in two previous examples, the OPE between
$W^{\lambda,\hat{A}=1}_{\mathrm{F},h_{1}=5}(z)$ and
$Q^{\lambda,\hat{B}=1}_{\mathrm{B},h_{2}+\frac{1}{2}=\frac{7}{2}}(w)$ where
$h_{1}=h_{2}+2$ can be obtained and the seventh order pole contains the
structure constant
$-\frac{6144}{35}(\lambda-1)(\lambda+1)(\lambda+2)(2\lambda-3)(2\lambda+1)(2\lambda+3)$
appearing in the current
$Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h=\frac{3}{2}}(w)$. On the other hand,
the structure constant $q_{\mathrm{F}}^{5,\frac{7}{2},5}(m,r,\lambda)$
contains the $\lambda$ dependent factor
$\frac{4}{1575}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(4\lambda^{2}-14\lambda-9)$.
By subtracting the contribution
$\frac{2}{225}(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)$
coming from the structure constant
$q_{\mathrm{F}}^{4,\frac{7}{2},5}(m,r,\lambda)$, where $4$ is realized by
$(h_{1}-1)$, from the above, we obtain
$-\frac{2}{525}(\lambda-1)(\lambda+1)(\lambda+2)(2\lambda-3)(2\lambda+1)(2\lambda+3)$.
Note that the additional term in the
$q_{\mathrm{F}}^{4,\frac{7}{2},5}(m,r,\lambda)$ contains the factor $\lambda$.
By considering the numerical factor $46080$ when we move from the modes in the
commutator to the differential operators in the OPE and multiplying this into
the above factor, we obtain the previous structure constant in the current
$Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}-h=\frac{3}{2}}(w)$. This implies that if
the weights $h_{1}$ and $h_{2}$ do not satisfy the above constraint
($h_{1}=h_{2}$ or $h_{1}=h_{2}+1$) for nonzero $\lambda$, then the formula in
the first equation of (3.18) cannot be used fully because the extra
contribution from the structure constant on the right hand sides of the
corresponding OPE occur. As before, for the poles less than the critical
singular term, still we can use some of terms in the corresponding expression
of the first relation of (3.18). Similarly, let us consider the last equation
of (3.18). Then the OPE between
$W^{\lambda,\hat{A}=1}_{\mathrm{B},h_{1}=5}(z)$ and
$\bar{Q}^{\lambda,\hat{B}=1}_{\mathrm{B},h_{2}+\frac{1}{2}=\frac{7}{2}}(w)$
provides the seventh order pole with
$-\frac{2048}{35}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(2\lambda+3)(11\lambda-10)$
appearing in the current $\bar{Q}^{\lambda}_{\frac{3}{2}}(w)$. This can be
obtained by adding the extra contribution from
$q_{\mathrm{B}}^{4,\frac{7}{2},5}(m,r,\lambda)$ in addition to the one from
$q_{\mathrm{B}}^{5,\frac{7}{2},5}(m,r,\lambda)$ as in previous paragraph. Note
that the $\bar{Q}^{\lambda}_{\frac{1}{2}}(w)$ term on the right hand side of
the OPE can be determined by the contribution from the structure constant
$q_{\mathrm{B}}^{5,\frac{7}{2},6}(m,r,\lambda)$ only. Therefore, the last
equation of (3.18) should be modified for nonzero $\lambda$ with more general
$h_{1}$ and $h_{2}$ where the weight $h_{2}$ is outside of the above allowed
region. The similar behaviors corresponding to the second and the third
equations of (3.18) we do not present in this paper can appear..

####  3.1.4 The final anticommutator relation with $h_{1}=h_{2}$ for nonzero
$\lambda$

The final anticommutator relation between the fermionic currents is summarized
by

$\displaystyle\\{(Q^{\lambda,\hat{A}}_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-1}_{h=0}\,q^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s,\lambda)\,\Bigg{(}i\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{F,h_{1}+h_{2}-h})_{r+s}$
(3.21) $\displaystyle+\\!$
$\displaystyle\,\delta^{\hat{A}\hat{B}}\,(W^{\lambda}_{F,h_{1}+h_{2}-h})_{r+s}\Bigg{)}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-1}_{h=0}\,q^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s,\lambda)\,\Bigg{(}-i\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{B,h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\,\delta^{\hat{A}\hat{B}}\,(W^{\lambda}_{B,h_{1}+h_{2}-h})_{r+s}\Bigg{)}$
$\displaystyle+\\!$
$\displaystyle\left(\begin{array}[]{c}r+h_{1}-\frac{1}{2}\\\ h_{1}+h_{2}\\\
\end{array}\right)\,c_{Q}(h_{1},h_{2},\lambda)\,\delta^{\hat{A}\hat{B}}\,q^{h_{1}+h_{2}-2}\delta_{r+s}\,.$

The corresponding OPEs for fixed $h_{1}$ and $h_{2}$ can be found in
Appendices (C.13) and (C.14) 555 As in the examples appearing in the previous
footnotes, the OPE between
$Q^{\lambda,\hat{A}=1}_{h_{1}+\frac{1}{2}=\frac{3}{2}}(z)$ and
$\bar{Q}^{\lambda,\hat{B}=1}_{h_{2}+\frac{1}{2}=\frac{7}{2}}(w)$ where
$h_{1}=h_{2}-2$ can be determined and the fourth order pole contains the
structure constant $\frac{128}{5}(\lambda-1)(2\lambda-3)(2\lambda-1)$
appearing in the current $W^{\lambda}_{F,h_{1}+h_{2}-h=1}(w)$. On the other
hand, the structure constant
$o_{\mathrm{F}}^{\frac{3}{2},\frac{7}{2},3}(r,s,\lambda)$ contains the
$\lambda$ dependent factor
$\frac{8}{15}(2\lambda-1)(4\lambda^{2}+2\lambda+3)$. We can take the sum (due
to the odd $h$) of these with proper numerical value $-48$ for the latter and
obtain $-\frac{128}{5}\lambda(2\lambda-1)(2\lambda+7)$ which vanishes at
$\lambda=0$ and this implies that this extra contribution should appear.
Furthermore, the fourth order pole contains the structure constant
$\frac{256}{5}\lambda(\lambda+1)(2\lambda+1)$ appearing in the current
$W^{\lambda}_{B,h_{1}+h_{2}-h=1}(w)$. On the other hand, the structure
constant $o_{\mathrm{B}}^{\frac{3}{2},\frac{7}{2},3}(r,s,\lambda)$ contains
$\frac{16}{15}\lambda(4\lambda^{2}-6\lambda+5)$. Obviously they are different
from each other and the extra term
$-\frac{256}{5}(\lambda-4)\lambda(2\lambda-1)$ should appear from similar
analysis. Similarly, the third order pole contains the structure constant
$\frac{96}{5}(\lambda-1)(2\lambda-3)$ appearing in the current
$W^{\lambda}_{F,h_{1}+h_{2}-h=2}(w)$. Moreover, the structure constant
$o_{\mathrm{F}}^{\frac{3}{2},\frac{7}{2},2}(r,s,\lambda)$ contains
$-\frac{4}{5}(4\lambda^{2}+10\lambda-9)$. By taking the difference (due to the
even $h$) between the latter with an additional numerical value $8$ and the
former, we obtain $-32\lambda(2\lambda-1)$ which vanishes at $\lambda=0$ and
should appear. The third order pole also contains the structure constant
$\frac{96}{5}(\lambda+1)(2\lambda+1)$ appearing in the current
$W^{\lambda}_{B,h_{1}+h_{2}-h=2}(w)$. On the other hand, the structure
constant $o_{\mathrm{B}}^{\frac{3}{2},\frac{7}{2},2}(r,s,\lambda)$ contains
$-\frac{4}{5}(4\lambda^{2}-14\lambda-3)$. We obtain $-32\lambda(2\lambda-1)$
from the difference between the latter and the former as before and this
$\lambda$ dependent contribution should appear. If the weights $h_{1}$ and
$h_{2}$ are not equal to each other for nonzero $\lambda$, then the formula in
the first equation of (3.21) cannot be used fully because the extra
contribution from the structure constant on the right hand sides of the
corresponding OPE arises. . The additional $(-1)^{h}$ factor appears when we
change the anticommutator relations into the corresponding OPEs. Because the
lowest weight of $W_{B,h}^{\lambda,\bar{a}b}$ is given by $1$, the upper limit
of the second summation is also given by $h=h_{1}+h_{2}-1$ which is the same
as the one in the first summation of (3.21). Here the central term appearing
in (3.21) can be described by

$\displaystyle c_{Q}(h_{1},h_{2},\lambda)$ $\displaystyle\equiv$
$\displaystyle
8(-4)^{h_{1}+h_{2}-4}\delta_{b\bar{a}}\delta_{d\bar{c}}\Bigg{[}-Q_{\lambda,\bar{a}b}^{(h_{1}+1)+}(z)Q_{\lambda,\bar{c}d}^{(h_{2}+1)+}(w)+Q_{\lambda,\bar{a}b}^{(h_{1}+1)-}(z)Q_{\lambda,\bar{c}d}^{(h_{2}+1)-}(w)$
(3.22) $\displaystyle-$ $\displaystyle
Q_{\lambda,\bar{a}b}^{(h_{1}+1)+}(z)\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)-}(w)+Q_{\lambda,\bar{a}b}^{(h_{1}+1)-}(z)\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)+}(w)$
$\displaystyle-$
$\displaystyle(-1)^{h_{1}+h_{2}}\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)+}(z)\,Q_{\lambda,\bar{a}b}^{(h_{1}+1)+}(w)+(-1)^{h_{1}+h_{2}}\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)-}(z)\,Q_{\lambda,\bar{a}b}^{(h_{1}+1)-}(w)$
$\displaystyle-$
$\displaystyle(-1)^{h_{1}+h_{2}}\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)+}(z)\,Q_{\lambda,\bar{a}b}^{(h_{1}+1)-}(w)$
$\displaystyle+$
$\displaystyle(-1)^{h_{1}+h_{2}}\,Q_{\lambda,\bar{c}d}^{(h_{2}+1)-}(z)\,Q_{\lambda,\bar{a}b}^{(h_{1}+1)+}(w)\Bigg{]}_{\frac{1}{(z-w)^{h_{1}+h_{2}+1}}}\,,$

where we can use Appendix (D.1).

Therefore, the seven (anti)commutator relations between the nonsinglet
currents are given by (3.4), (3.16), (3.18) and (3.21).

###  3.2 The (anti)commutator relations between the nonsinglet currents in
explicit forms with $h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$

In order to construct the algebra from the ${\cal N}=4$ multiplets, we should
rewrite the previous (anti)commutator relations in the basis of four
components of $(\bar{a}b)=(11,12,21,22)$ in each current. For example, from
(3.4), the commutator relation between $\hat{A}=1$ (sum of $(12)$ and $(21)$)
and $\hat{B}=1$ is known. Moreover, the commutator relation between
$\hat{A}=1$ and $\hat{B}=2$ (difference of $(12)$ and $(21)$ up to an overall
factor) is known. Then we obtain the commutator relation between $\hat{A}=1$
and the element $(12)$ current by adding the above two commutator relations.
By realizing that there is no singular term in the commutator relation between
the element $(12)$ current and itself, the above analysis leads to the
commutator relation between the element $(21)$ and the element $(12)$ as
follows:

$\displaystyle\big{[}(W^{\lambda,21}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,12}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,11}_{\mathrm{F},h_{1}+h_{2}-2-h}+W^{\lambda,22}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{2}\,\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{F}(h_{1},h_{2},\lambda)\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}$
$\displaystyle-\\!$
$\displaystyle\frac{1}{2}\,\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,11}_{\mathrm{F},h_{1}+h_{2}-2-h}-W^{\lambda,22}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,.$

On the other hand, by subtracting the previous two commutator relations and by
using the fact that there is no singular term in the commutator relation
between the element $(21)$ current and itself, we obtain the following
commutator relation between the element $(12)$ and the element $(21)$ as
follows:

$\displaystyle\big{[}(W^{\lambda,12}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,21}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,11}_{\mathrm{F},h_{1}+h_{2}-2-h}+W^{\lambda,22}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{2}\,\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{F}(h_{1},h_{2},\lambda)\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}$
$\displaystyle-\\!$
$\displaystyle\frac{1}{2}\,\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,11}_{\mathrm{F},h_{1}+h_{2}-2-h}-W^{\lambda,22}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,.$

Similarly, the commutator relation having both $\hat{A}=1$ and $\hat{B}=3$ and
the commutator relation having $\hat{A}=2$ and $\hat{B}=3$ provide the
commutator relation between the element $(12)$ and the current having
$\hat{B}=3$ and the commutator relation between the element $(21)$ and the
current having $\hat{B}=3$. Furthermore, the commutator relation having both
$\hat{A}=1$ and $\hat{B}=0$ and the commutator relation having $\hat{A}=2$ and
$\hat{B}=0$ provide the commutator relation between the element $(12)$ and the
current having $\hat{B}=0$ and the commutator relation between the element
$(21)$ and the current having $\hat{B}=0$. Then we obtain the following
commutator relations by linear combination of previous four commutators

$\displaystyle\big{[}(W^{\lambda,12}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,11}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,12}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,12}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,22}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}+\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,12}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,21}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,11}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}+\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,21}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,21}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,22}_{\mathrm{F},h_{2}})_{n}\big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})q^{h}p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)(W^{\lambda,21}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}.$
(3.29)

From the results of the commutator relation between the current having the
index $\hat{A}=0$ and the current having $\hat{B}=1$ and the commutator
relation between the current having the index $\hat{A}=0$ and the current
having $\hat{B}=2$, the commutator relation between the current having the
index $\hat{A}=0$ and the element $(12)$ and the one between the current
having the index $\hat{A}=0$ and the element $(21)$ can be determined.
Moreover, from the commutator relation having the index $\hat{A}=3$ and
$\hat{B}=1$ and the commutator relation having the index $\hat{A}=3$ and
$\hat{B}=2$, the commutator relation between the current having the index
$\hat{A}=3$ and the element $(12)$ and the one between the current having the
index $\hat{A}=3$ and the element $(21)$ can be fixed. Therefore, by linear
combinations of these results, we can determine the following commutator
relations

$\displaystyle\big{[}(W^{\lambda,11}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,12}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}+\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,12}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,22}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,12}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,12}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,11}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,21}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,21}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,22}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,21}_{\mathrm{F},h_{2}})_{n}\big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}+\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}})q^{h}p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)(W^{\lambda,21}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}.$
(3.30)

Finally, from the results of the commutator relation between the current
having the index $\hat{A}=0$ and the current having the index $\hat{B}=3$ and
the commutator relation between the current having the index $\hat{A}=0$ and
the current having the index $\hat{B}=0$, we can determine the following
commutator relations by realizing that there is no nontrivial commutator
relation between the element $(11)$ and the element $(22)$

$\displaystyle\big{[}(W^{\lambda,11}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,11}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,11}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,$
(3.33) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{F}(h_{1},h_{2},\lambda)\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}\,,$
$\displaystyle\big{[}(W^{\lambda,22}_{\mathrm{F},h_{1}})_{m},(W^{\lambda,22}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)\,(W^{\lambda,22}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\,$
(3.36) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\
\end{array}\right)\,c_{F}(h_{1},h_{2},\lambda)\,q^{h_{1}+h_{2}-4}\,\delta_{m+n}\,.$

Therefore, there are (3.2), (3.2), (3.29), (3.30) and (3.36). Other complete
relations are summarized by Appendices (E.12), (E.13), (E.14), (E.15), (E.16)
and (E.28). By adding the two equations of (3.36), we obtain the commutator
between the singlet currents and by taking the contractions of the currents
with the vanishing $q$ limit for $\lambda=0$ we obtain the $w_{1+\infty}$
algebra [21, 22, 7].

##  4  The ${\cal N}=4$ supersymmetric linear $W_{\infty}^{2,2}$ algebra
between the ${\cal N}=4$ multiplets

Based on the results in previous sections, we present the first five
commutator relations between the ${\cal N}=4$ multiplets. In the footnotes,
some particular examples for the specific weights $h_{1}$ and $h_{2}$ show
that there are extra terms on the right hand sides of the commutators for the
nonzero $\lambda$ we explained in the introduction.

###  4.1 The commutator relation between the lowest components with
$h_{1}=h_{2},h_{2}\pm 1$ for nonzero $\lambda$

Let us consider the commutator relation between the lowest components
$\Phi_{0}^{(h_{1})}(z)$ and $\Phi_{0}^{(h_{2})}(w)$ of the ${\cal N}=4$
multiplet. Because there are no singular terms in the OPE of
$W_{F,h_{1}}^{\lambda,\bar{a}a}(z)$ and $W_{B,h_{2}}^{\lambda,\bar{b}b}(w)$
according to (2.1), the commutator relation consists of two parts. That is,
they are given by
$\big{[}(W_{F,h_{1}}^{\lambda,\bar{a}a})_{m},(W_{F,h_{2}}^{\lambda,\bar{b}b})_{n}\big{]}$
and
$\big{[}(W_{B,h_{1}}^{\lambda,\bar{a}a})_{m},(W_{B,h_{2}}^{\lambda,\bar{b}b})_{n}\big{]}$.
Then we can use the relation (3.36) and the last two relations of Appendix
(E.12). We need to express the right hand sides of these commutator relations
in terms of the components of the ${\cal N}=4$ multiplet in order to present
the complete algebra between them. The two relations in (3.36) and the last
two relations of Appendix (E.12) can be added respectively because the
structure constants are common at each expression.

As described before, from the relations (2.15) and (2.19), we obtain the
following relations

$\displaystyle W_{F,h}^{\lambda,\bar{a}a}$ $\displaystyle=$
$\displaystyle-\frac{1}{(-4)^{h-2}}\,\Phi_{0}^{(h)}-\frac{(h-1+2\lambda)}{8(2h-1)(-4)^{h-6}}\,\tilde{\Phi}_{2}^{(h-2)}\,,$
$\displaystyle W_{B,h}^{\lambda,\bar{a}a}$ $\displaystyle=$
$\displaystyle\frac{1}{(-4)^{h-2}}\,\Phi_{0}^{(h)}-\frac{(h-2\lambda)}{8(2h-1)(-4)^{h-6}}\,\tilde{\Phi}_{2}^{(h-2)}\,.$
(4.1)

Once we observe the currents on the left hand sides, then we should rewrite
them in terms of the components of ${\cal N}=4$ multiplet according to (4.1).

On the other hand, there exist the following relations between the lowest
bosonic currents and the lowest component $\Phi_{0}^{(1)}$ of the first ${\cal
N}=4$ multiplet from (2.15) and the weight-$1$ current
$U=2(W_{F,1}^{\lambda,\bar{a}a}+W_{B,1}^{\lambda,\bar{a}a})$ of the ${\cal
N}=4$ stress energy tensor

$\displaystyle W_{F,1}^{\lambda,\bar{a}a}$ $\displaystyle=$ $\displaystyle
4\,\Phi_{0}^{(1)}+\lambda\,U\,,$ $\displaystyle W_{B,1}^{\lambda,\bar{a}a}$
$\displaystyle=$
$\displaystyle-4\,\Phi_{0}^{(1)}+\frac{1}{2}\,(1-2\lambda)\,U\,.$ (4.2)

This implies that by comparing (4.1) with (4.2), we can identify the previous
weight-$1$ current as $\tilde{\Phi}_{2}^{(-1)}\equiv 256\,U$.

Furthermore, there are relations between the lowest component $\Phi_{0}^{(2)}$
of the second ${\cal N}=4$ multiplet and the weight-$2$ stress energy tensor
$L$ of the ${\cal N}=4$ stress energy tensor together with (2.15) and (2.13)
as follows:

$\displaystyle W_{F,2}^{\lambda,\bar{a}a}$ $\displaystyle=$
$\displaystyle-\Phi_{0}^{(2)}+\frac{1}{3}\,(1+2\lambda)\,L\,,$ $\displaystyle
W_{B,2}^{\lambda,\bar{a}a}$ $\displaystyle=$
$\displaystyle\Phi_{0}^{(2)}+\frac{2}{3}\,(1-\lambda)\,L\,.$ (4.3)

Again, from the relations (4.1) and (4.3), we identify the stress energy
tensor in terms of the component of ${\cal N}=4$ multiplet as
$\tilde{\Phi}_{2}^{(0)}\equiv-32\,L$.

Therefore, the commutator relation between the lowest component of the
$h_{1}$-th ${\cal N}=4$ multiplet and the lowest component of the $h_{2}$-th
${\cal N}=4$ multiplet can be described as, by substituting the relation (4.1)
into the above right hand sides of the relevant commutator relations described
before,

$\displaystyle\big{[}(\Phi^{(h_{1})}_{0})_{m},(\Phi^{(h_{2})}_{0})_{n}\big{]}=\frac{(-4)^{h_{1}-2}}{(2h_{1}-1)}\,\frac{(-4)^{h_{2}-2}}{(2h_{2}-1)}\Bigg{[}\left(\begin{array}[]{c}m+h_{1}-1\\\
h_{1}+h_{2}-1\\\ \end{array}\right)\,q^{h_{1}+h_{2}-4}$ (4.6)
$\displaystyle\times\Bigg{(}(h_{1}-2\lambda)(h_{2}-2\lambda)\,c_{F}(h_{1},h_{2},\lambda)+(h_{1}-1+2\lambda)(h_{2}-1+2\lambda)\,c_{B}(h_{1},h_{2},\lambda)\Bigg{)}\,\delta_{m+n}$
$\displaystyle+\sum_{h=0,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-3}\,\Bigg{(}-(h_{1}-2\lambda)(h_{2}-2\lambda)\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)(h_{2}-1+2\lambda)\,q^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n,\lambda)\,\Bigg{)}\,\frac{1}{(-4)^{h_{1}+h_{2}-h-4}}\,(\Phi_{0}^{(h_{1}+h_{2}-2-h)})_{m+n}$
$\displaystyle-\sum_{h=0,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-3}\,\Bigg{(}(h_{1}-2\lambda)(h_{2}-2\lambda)(h_{1}+h_{2}-h-3+2\lambda)\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)(h_{2}-1+2\lambda)(h_{1}+h_{2}-h-2-2\lambda)\,q^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n,\lambda)\,\Bigg{)}$
$\displaystyle\times\frac{1}{8(2h_{1}+2h_{2}-2h-5)(-4)^{h_{1}+h_{2}-h-8}}\,(\tilde{\Phi}_{2}^{(h_{1}+h_{2}-h-4)})_{m+n}\Bigg{]}\,.$
(4.7)

Each central term is given by (3.12) and (3.17). Due to the even weight $h$ in
the summation, the currents of even (or odd) weights can occur depending on
the weights $h_{1}$ and $h_{2}$. When we change the above commutator relation
(4.7) into the corresponding OPE, due to the factor $(-1)^{h-1}$ for even $h$,
then there exists an extra minus sign on the right hand side of the OPE. We
can easily observe that for the maximum value of dummy variable $h$,
$h=h_{1}+h_{2}-3$ with odd $(h_{1}+h_{2})$, on the right hand side of (4.7),
there appear the currents $\Phi_{0}^{(1)}$ and $\tilde{\Phi}_{2}^{-1}$ which
is related to the previous current $U$ of ${\cal N}=4$ stress energy tensor.
Note that we have vanishing structure constant
$p_{B}^{h_{1},h_{2},h_{1}+h_{2}-3}$ at $\lambda=0$ [8]. On the other hand, for
even $(h_{1}+h_{2})$, the maximum value of the weight $h$ is given by
$h=h_{1}+h_{2}-4$ because the weight $h$ should be even. For this value, there
appear the currents $\Phi_{0}^{(2)}$ and $\tilde{\Phi}_{2}^{(0)}$, which is
related to the previous current $L$, with proper $\lambda$ dependent structure
constants on the right hand side of (4.7). Let us emphasize that for nonzero
$\lambda$, the above commutator holds for the arbitrary weight $h_{1}$ under
the condition $h_{1}=h_{2}$ or $h_{1}=h_{2}\pm 1$. We take the particular
example which shows that if the weights $h_{1}$ and $h_{2}$ do not satisfy
this condition, then there exist other terms on the right hand of the above
commutator 666 Let us take the OPE between $\Phi^{(h_{1}=1)}_{0}(z)$ and
$\Phi^{(h_{2}=3)}_{0}(w)$ where the weights satisfy $h_{1}=h_{2}-2$. The third
order pole of this OPE provides the structure constant
$2048\lambda(2\lambda-1)$ appearing in the current $\Phi^{(1)}_{0}(w)$ and the
structure constant $\frac{4}{5}\lambda(2\lambda-1)(4\lambda-1)$ appearing in
the current $U(w)$. According to (4.7), the exponent of the first current
implies $h_{1}+h_{2}-2-h=2-h$. Moreover, $h$ is given by $h=0$ from the
summation and the exponent is $2$. Therefore, the current $\Phi^{(1)}_{0}(w)$
cannot appear from the (4.7). Furthermore, the exponent in the second current
implies $h_{1}+h_{2}-4-h=-h=-1$ from the presence of $U$. Moreover, $h$ is
given by $h=0$ from the summation and there is a contradiction. In this case
also, the current $U(w)$ cannot appear from the corresponding term in (4.7).
Note that the two structure constants contain the $\lambda$ factor and they
become zero at $\lambda=0$. From the above analysis, there exist extra terms
on the right hand side of the commutator (or corresponding OPE) for the
weights which do not satisfy the constraint $h_{1}=h_{2}$ or $h_{1}=h_{2}\pm
1$ we mentioned before. As in previous footnotes in section $3$, this feature
also arises for the singlet currents. .

###  4.2 The commutator relation between the lowest component and the second
component with $h_{1}=h_{2},h_{2}+1$ for nonzero $\lambda$

Let us consider the commutator relation between the lowest component
$\Phi_{0}^{(h_{1})}(z)$ of the $h_{1}$-th ${\cal N}=4$ multiplet in (2.15) and
the second component $\Phi_{\frac{1}{2}}^{(h_{2}),i}(w)$ of $h_{2}$-th ${\cal
N}=4$ multiplet in (2.16).

We expect to have the fermionic currents
$Q_{h+\frac{1}{2}}^{\lambda,\bar{a}b}$ and
$\bar{Q}_{h+\frac{1}{2}}^{\lambda,b\bar{a}}$ on the right hand side of the
commutator relation because the product of bosonic and fermionic operators
produce the fermionic ones. As before, we need to rewrite them in terms of the
components of the ${\cal N}=4$ multiplet in order to complete the algebra we
are considering. For the index $i=1$ of these components of the ${\cal N}=4$
multiplets, the following relations are satisfied

$\displaystyle\frac{1}{2}\,\Big{(}Q^{\lambda,11}_{h+\frac{1}{2}}+i\sqrt{2}\,Q^{\lambda,12}_{h+\frac{1}{2}}+2i\sqrt{2}\,Q^{\lambda,21}_{h+\frac{1}{2}}-2\,Q^{\lambda,22}_{h+\frac{1}{2}}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\Bigg{[}\frac{1}{4(-4)^{h-4}}\,\Phi^{(h),1}_{\frac{1}{2}}-\frac{1}{4(-4)^{h-5}}\,\tilde{\Phi}^{(h-1),1}_{\frac{3}{2}}\Bigg{]}\,,$
$\displaystyle\frac{1}{2}\Big{(}2\,\bar{Q}^{\lambda,11}_{h+\frac{1}{2}}+2i\sqrt{2}\bar{Q}^{\lambda,12}_{h+\frac{1}{2}}+i\sqrt{2}\bar{Q}^{\lambda,21}_{h+\frac{1}{2}}-\bar{Q}^{\lambda,22}_{h+\frac{1}{2}}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\frac{1}{4(-4)^{h-4}}\Phi^{(h),1}_{\frac{1}{2}}+\frac{1}{4(-4)^{h-5}}\tilde{\Phi}^{(h-1),1}_{\frac{3}{2}}\Bigg{]}.$

Note that the weight on both sides is given by $(h+\frac{1}{2})$. On the right
hand side, the $(h-1)$-th component of ${\cal N}=4$ multiplet appears also.
Other relations for $i=2,3,4$ appear in Appendix (F.1). For $h=0$, the above
relation (LABEL:twotwo) with others in Appendix (F.1) implies that there
exists the relation between the currents
$\tilde{\Phi}^{(-1),i}_{\frac{3}{2}}=-\frac{1}{4}\,\Phi^{(0),i}_{\frac{1}{2}}$
together with (2.16) and (2.18) because the left hand side of the first
relation of (LABEL:twotwo) is identically zero and moreover, the
weight-$\frac{1}{2}$ current of the ${\cal N}=4$ stress energy tensor has the
following relation $-i\,\Gamma^{i}=64\,\Phi^{(0),i}_{\frac{1}{2}}$ with
(2.16). For $h=1$, the corresponding weight-$\frac{3}{2}$ currents of ${\cal
N}=4$ stress energy tensor can be written as
$G^{i}=64\,\tilde{\Phi}^{(0),i}_{\frac{3}{2}}$ with (2.18) by subtracting the
two relation of (LABEL:twotwo). Therefore, the components,
$U,\Gamma^{i},G^{i}$ and $L$ of ${\cal N}=4$ stress energy tensor, except the
weight-$1$ currents $T^{ij}$ which will appear in next subsection, are given
by the components of the ${\cal N}=4$ multiplet,
$\tilde{\Phi}_{2}^{(-1)},\Phi_{\frac{1}{2}}^{(0),i},\tilde{\Phi}_{\frac{3}{2}}^{(0),i}$
and $\tilde{\Phi}_{2}^{(0)}$ up to normalization respectively.

The result of the commutator relation we are considering, by replacing all the
fermionic currents with the ${\cal N}=4$ components appearing in
(LABEL:twotwo) and Appendix (F.1), can be summarized by

$\displaystyle\big{[}(\Phi^{(h_{1})}_{0})_{m},(\Phi^{(h_{2}),i}_{\frac{1}{2}})_{r}\big{]}=5\frac{(-4)^{h_{1}-2}}{(2h_{1}-1)}\,4(-4)^{h_{2}-4}\Bigg{[}\Bigg{(}-(h_{1}-2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-3}\,q^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)$
$\displaystyle-(h_{1}-2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-2}q^{h}\,(-1)^{h}q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)+(h_{1}-1+2\lambda)\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-2}\,q^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\Bigg{)}\,\frac{1}{8(-4)^{h_{1}+h_{2}-h-6}}\,(\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h)})_{m+r}$
$\displaystyle+\Bigg{(}\sum_{h=-1}^{h_{1}+h_{2}-3}\,(h_{1}-2\lambda)\,q^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)-\sum_{h=-1}^{h_{1}+h_{2}-2}\,(h_{1}-2\lambda)\,q^{h}\,(-1)^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)$
$\displaystyle-\sum_{h=-1}^{h_{1}+h_{2}-3}\,(h_{1}-1+2\lambda)\,q^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)$
(4.9)
$\displaystyle+\sum_{h=-1}^{h_{1}+h_{2}-2}\,(h_{1}-1+2\lambda)\,q^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,\lambda)\Bigg{)}\,\frac{1}{8(-4)^{h_{1}+h_{2}-h-7}}\,(\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-3-h)})_{m+r}\,\Bigg{]}\,.$

Note that the upper limit of $h$ having the term with the factor $(-1)^{h}$ is
given by $h=h_{1}+h_{2}-2$. Except these four terms, the range of $h$ in all
the other terms is the same. Then these four terms occur if their structure
constants are nonvanishing. Depending on the odd $h$ or even $h$, some of
terms in (4.9) can cancel each other because four kinds of each two terms have
the same structure except the factor $(-1)^{h}$. At $h=h_{1}+h_{2}-2$, there
appear the current $\Phi_{\frac{1}{2}}^{(0),i}$ and the current
$\tilde{\Phi}_{\frac{3}{2}}^{(-1),i}$ which are related to the previous
weight-$\frac{1}{2}$ current $\Gamma^{i}$ as above, with the relevant
structure constants. Moreover, we have vanishing structure constants
$q_{F}^{h_{1},h_{2}+\frac{1}{2},h_{1}+h_{2}-2}=0=q_{B}^{h_{1},h_{2}+\frac{1}{2},h_{1}+h_{2}-2}$
at $\lambda=0$ [8]. For the $h\leq h_{1}+h_{2}-3$ with odd $h$ in (4.9), there
appear the current $\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-3-h),i}$ terms
because the coefficients of the current
$\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h),i}$ terms are identically vanishing.
Note that there are relative signs in (4.9). On the other hand, for the $h\leq
h_{1}+h_{2}-3$ with even $h$, there appear the current
$\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h),i}$ terms because the coefficients of
the current $\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-3-h),i}$ terms are
identically zero. We also present some example where the weights $h_{1}$ and
$h_{2}$ do not satisfy the condition $h_{1}=h_{2}$ or $h_{1}=h_{2}+1$ 777Along
the line of the footnote 6, we consider the OPE between
$\Phi^{(h_{1}=4)}_{0}(z)$ and $\Phi^{(h_{2}=2),i}_{\frac{1}{2}}(w)$ where
$h_{1}=h_{2}+2$. The sixth order pole of this OPE gives us the structure
constant
$\frac{2048}{7}(\lambda-1)\lambda(2\lambda-1)(2\lambda+1)(4\lambda-1)$
appearing in the current $\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h=0),i}(w)$ with
the weight $h=4$. By substituting the various expressions in the corresponding
terms of (4.9), we can check that we obtain the above structure constant
correctly where $q_{F}^{4,\frac{5}{2},4}$ term corresponds to
$\frac{256}{3}(\lambda-1)\lambda(2\lambda-3)(2\lambda-1)(2\lambda+1)$ while
$q_{B}^{4,\frac{5}{2},4}$ term corresponds to
$-\frac{512}{3}(\lambda-1)\lambda(\lambda+1)(2\lambda-1)(2\lambda+1)$. There
is no current $\tilde{\Phi}_{\frac{3}{2}}^{(-1),i}(w)$ term in the sixth order
pole from our calculation and this is consistent with (4.9) because all the
coefficients vanish for the even $h=4$. Now we can move to the fifth order
pole of this OPE. The current $G^{i}(w)$ term of ${\cal N}=4$ stress energy
tensor has the structure constant
$-\frac{512}{7}(\lambda-1)(2\lambda+1)(6\lambda^{2}-3\lambda-4)$ which can be
identified with the coefficient appearing in the second current
$\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-3-h=0),i}(w)$ of (4.9) by
substituting $q_{F}^{4,\frac{5}{2},3}$ and $q_{B}^{4,\frac{5}{2},3}$ correctly
as before. On the other hand, there exists the current
$\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h=1),i}(w)$ term with the structure
constant $\frac{8192}{3}(\lambda-1)\lambda(2\lambda-1)(2\lambda+1)$ having the
$\lambda$ factor. We can check that this structure constant is equal to the
one in the first current terms in (4.9) where the previous
$q_{F}^{4,\frac{5}{2},3}$ and $q_{B}^{4,\frac{5}{2},3}$ are replaced by
$q_{F}^{3,\frac{5}{2},3}$ and $q_{B}^{3,\frac{5}{2},3}$ respectively. That is,
the $h_{1}$ is replaced by $(h_{1}-1)$. Therefore, we cannot use the equation
(4.9) for this particular pole fully. We can check that the remaining lower
order poles can be described by (4.9) without any extra terms precisely..

###  4.3 The commutator relation between the lowest component and the third
component with $h_{1}=h_{2},h_{2}+1,h_{2}+2$ for nonzero $\lambda$

Let us consider the commutator relation between the lowest component
$\Phi_{0}^{(h_{1})}(z)$ of the $h_{1}$-th ${\cal N}=4$ multiplet and the third
component $\Phi_{1}^{(h_{2}),ij}(w)$ of the $h_{2}$-th ${\cal N}=4$ multiplet.
Because there are no singular terms in the OPE of
$W_{F,h_{1}}^{\lambda,\bar{a}b}(z)$ and $W_{B,h_{2}}^{\lambda,\bar{c}d}(w)$,
the commutator relation consists of two parts. That is, there are
$\big{[}(W_{F,h_{1}}^{\lambda,\bar{a}a})_{m},(W_{F,h_{2}}^{\lambda,\bar{c}d})_{n}\big{]}$
and
$\big{[}(W_{B,h_{1}}^{\lambda,\bar{a}a})_{m},(W_{B,h_{2}}^{\lambda,\bar{c}d})_{n}\big{]}$.
Then we can use the previous relations, (3.2), (3.2), (3.29), (3.30) and
(3.36) and Appendix (E.12). We need to express the right hand sides of these
commutator relations in terms of the components of the ${\cal N}=4$ multiplet
as before.

We obtain the following relations

$\displaystyle
2i\,W^{\lambda,11}_{\mathrm{B},h+1}-\sqrt{2}\,W^{\lambda,12}_{\mathrm{B},h+1}-2i\,\,W^{\lambda,22}_{\mathrm{B},h+1}$
$\displaystyle=$
$\displaystyle\frac{1}{8(-4)^{h-4}}\Bigg{[}\Phi_{1}^{(h),12}-\Phi_{1}^{(h),34}\Bigg{]}\,,$
$\displaystyle
2i\,W^{\lambda,11}_{\mathrm{F},h+1}-2\sqrt{2}\,W^{\lambda,12}_{\mathrm{F},h+1}-2i\,W^{\lambda,22}_{\mathrm{F},h+1}$
$\displaystyle=$
$\displaystyle\frac{1}{8(-4)^{h-4}}\Bigg{[}\Phi_{1}^{(h),12}+\Phi_{1}^{(h),34}\Bigg{]}\,.$
(4.10)

For other relations, we present them in Appendix (F.2). By using (4.10) and
Appendix (F.2), we can express the weight-$1$ current $T^{ij}$ of the ${\cal
N}=4$ stress energy tensor as $i\,T^{ij}=64\,\Phi_{1}^{(0),ij}$ with (2.17).
On the right hand sides of (4.10), there are only the components of $h$-th
${\cal N}=4$ multiplet. By linear combinations, any component of the $h$-th
${\cal N}=4$ multiplet can be written in terms of
$W_{F,h+1}^{\lambda,\bar{a}b}$ and $W_{B,h+1}^{\lambda,\bar{c}d}$ explicitly.

It turns out that the corresponding commutator relation, after substituting
all the bosonic currents in terms of the components of the ${\cal N}=4$
multiplet described above, can be written as

$\displaystyle\big{[}(\Phi^{(h_{1})}_{0})_{m},(\Phi^{(h_{2}),ij}_{1})_{n}\big{]}=\frac{(-4)^{h_{1}-2}}{(2h_{1}-1)}\,4(-4)^{h_{2}-4}\Bigg{[}-(h_{1}-2\lambda)\,\sum_{h=-1,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-2}\,q^{h}\,p_{\mathrm{F}}^{h_{1},h_{2}+1,h}(m,n,\lambda)$
$\displaystyle\times\frac{1}{8(-4)^{h_{1}+h_{2}-h-6}}\,\Bigg{(}\Phi_{1}^{(h_{1}+h_{2}-2-h),ij}+\frac{1}{2}\,\varepsilon^{ijkl}\,\Phi_{1}^{(h_{1}+h_{2}-2-h),kl}\Bigg{)}_{m+n}$
$\displaystyle+(h_{1}-1+2\lambda)\,\sum_{h=-1,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-2}\,q^{h}\,p_{\mathrm{B}}^{h_{1},h_{2}+1,h}(m,n,\lambda)$
$\displaystyle\times\frac{1}{8(-4)^{h_{1}+h_{2}-h-6}}\,\Bigg{(}\Phi_{1}^{(h_{1}+h_{2}-2-h),ij}-\frac{1}{2}\,\varepsilon^{ijkl}\,\Phi_{1}^{(h_{1}+h_{2}-2-h),kl}\Bigg{)}_{m+n}\,\Bigg{]}\,.$
(4.11)

Due to the even weight $h$, on the right hand side of (4.11), the currents of
odd (or even) weights can appear depending on the weights $h_{1}$ and $h_{2}$.
Due to the $SO(4)$ index $ij$ appearing on the left hand side of (4.11), the
field contents on the right hand side are different from the ones in (4.7). At
the maximum value of $h$, $h=h_{1}+h_{2}-2$, the current $\Phi_{1}^{(0),ij}$
term and the current $\varepsilon^{ijkl}\,\Phi_{1}^{(0),kl}$ term, which are
related to the previous weight-$1$ current of the ${\cal N}=4$ stress energy
tensor, occur on the right hand side of (4.11). Note that as before we have
vanishing structure constant $p_{B}^{h_{1},h_{2}+1,h_{1}+h_{2}-2}$ at
$\lambda=0$ [8]. In this case also, there some constraints on the weight
$h_{2}$ for nonzero $\lambda$ 888For the OPE between $\Phi^{(h_{1}=4)}_{0}(z)$
and $\Phi^{(h_{2}=1),ij}_{1}(w)$ where the weights satisfy $h_{1}=h_{2}+3$,
the fifth order pole of this OPE has the structure constant
$-\frac{8192}{7}\lambda(2\lambda-1)(4\lambda^{2}-2\lambda-5)$ in the current
$\Phi_{1}^{(0),ij}(w)$ term and the structure constant
$\frac{8192}{7}\lambda(2\lambda-1)(4\lambda-1)$ in the current
$\frac{1}{2}\,\varepsilon^{ijkl}\,\Phi_{1}^{(0),kl}(w)$ term from our
calculation. However, the corresponding structure constants
$p_{F}^{h_{1}=4,h_{2}+1=2,h=2}$ and $p_{B}^{h_{1}=4,h_{2}+1=2,h=2}$ do not
produce these $\lambda$ dependent structure constants respectively. This
implies that there are extra contributions in the structure constants for the
weights which do not satisfy the condition $h_{1}=h_{2}$, $h_{1}=h_{2}+1$, or
$h_{1}=h_{2}+2$..

###  4.4 The commutator relation between the lowest component and the fourth
component with $h_{1}=h_{2}+1,h_{2}+2$ for nonzero $\lambda$

In this case, we can use the previous relation (LABEL:twotwo) and Appendix
(F.1) by simply substituting $h$ with $(h+1)$ in order to rewrite the right
hand side of this commutator relation in terms of the components of the ${\cal
N}=4$ multiplet.

Then we can determine the following result for the commutator relation between
the lowest component $\Phi_{0}^{(h_{1})}(z)$ of the $h_{1}$-th ${\cal N}=4$
multiplet and the fourth component (and derivative term)
$\tilde{\Phi}_{\frac{3}{2}}^{(h_{2}),i}(w)$ of the $h_{2}$-th ${\cal N}=4$
multiplet

$\displaystyle\big{[}(\Phi^{(h_{1})}_{0})_{m},(\tilde{\Phi}^{(h_{2}),i}_{\frac{3}{2}})_{r}\big{]}=\frac{(-4)^{h_{1}-2}}{(2h_{1}-1)}\,4(-4)^{h_{2}-4}\Bigg{[}\Bigg{(}(h_{1}-2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-2}\,q^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)$
$\displaystyle-(h_{1}-2\lambda)\sum_{h=-1}^{h_{1}+h_{2}-1}q^{h}\,(-1)^{h}q_{\mathrm{F}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)-(h_{1}-1+2\lambda)\sum_{h=-1}^{h_{1}+h_{2}-2}q^{h}q_{\mathrm{B}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-1}\,q^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)\Bigg{)}\,\frac{1}{8(-4)^{h_{1}+h_{2}-h-5}}\,(\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-1-h),i})_{m+r}$
$\displaystyle+\Bigg{(}-(h_{1}-2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-2}\,q^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)$
(4.12)
$\displaystyle-(h_{1}-2\lambda)\sum_{h=-1}^{h_{1}+h_{2}-1}q^{h}\,(-1)^{h}q_{\mathrm{F}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)+(h_{1}-1+2\lambda)\sum_{h=-1}^{h_{1}+h_{2}-2}q^{h}q_{\mathrm{B}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)\,\sum_{h=-1}^{h_{1}+h_{2}-1}\,q^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{3}{2},h}(m,r,\lambda)\Bigg{)}\,\frac{1}{8(-4)^{h_{1}+h_{2}-h-6}}\,(\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-2-h),i})_{m+r}\,\Bigg{]}\,.$

The form of this (4.12) looks similar to the previous one in (4.9) because the
field contents are the same. The relative signs are different from each other
and we observe that after replacing $h_{2}$ with $(h_{2}+1)$ appearing in
(4.9), the corresponding expressions occur in (4.12). At $h=h_{1}+h_{2}-1$,
there appear the $\Phi_{\frac{1}{2}}^{(0),i}$ and the
$\tilde{\Phi}_{\frac{3}{2}}^{(-1),i}$ which are related to the
weight-$\frac{1}{2}$ current $\Gamma^{i}$. Except four terms having the factor
$(-1)^{h}$ with $h=h_{1}+h_{2}-1$, there some cancellation between the
currents. Furthermore, we have vanishing structure constants
$q_{F}^{h_{1},h_{2}+\frac{3}{2},h_{1}+h_{2}-1}=0=q_{B}^{h_{1},h_{2}+\frac{3}{2},h_{1}+h_{2}-1}$
at $\lambda=0$ [8]. For the $h\leq h_{1}+h_{2}-2$ with odd $h$ in (4.12),
there appear the current $\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-1-h),i}$ terms
because the coefficients of the current
$\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-2-h),i}$ terms are identically
vanishing. On the other hand, for the $h\leq h_{1}+h_{2}-2$ with even $h$,
there appear the current $\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-2-h),i}$
terms because the coefficients of the current
$\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h),i}$ terms are identically zero. This
behavior is different from the one in (4.9) because the appearance of signs
behaves differently. We also comment on the possible ranges for the weight
$h_{2}$ as described before 999 Similar to the previous examples, we consider
the OPE between $\Phi^{(h_{1}=4)}_{0}(z)$ and
$\Phi^{(h_{2}=1),i}_{\frac{3}{2}}(w)$ where the weights satisfy
$h_{1}=h_{2}+3$. The sixth order pole of this OPE provides the structure
constant
$\frac{32768}{7}(\lambda-1)\lambda(2\lambda-1)(2\lambda+1)(4\lambda-1)$
appearing in the current
$\tilde{\Phi}_{\frac{3}{2}}^{(h_{1}+h_{2}-2-h=-1),i}(w)$ with the weight
$h=4$. By substituting the various expressions in the corresponding terms of
(4.12), we can check that we obtain the above structure constant correctly
where $q_{F}^{4,\frac{5}{2},4}$ term corresponds to the $\lambda$ dependent
factor $\frac{256}{3}(\lambda-1)\lambda(2\lambda-3)(2\lambda-1)(2\lambda+1)$
while $q_{B}^{4,\frac{5}{2},4}$ term corresponds to the $\lambda$ dependent
factor $-\frac{512}{3}(\lambda-1)\lambda(\lambda+1)(2\lambda-1)(2\lambda+1)$.
There is no $\Phi_{\frac{1}{2}}^{(1),i}(w)$ term in the sixth order pole from
our calculation and this is consistent with (4.12) because all the
coefficients vanish for the even weight $h=4$. Now we can move to the fifth
order pole of above OPE. The $\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h=1),i}(w)$
term has the structure constant
$-\frac{2048}{7}(\lambda-1)(2\lambda+1)(6\lambda^{2}-3\lambda-4)$ which can be
identified with the coefficient appearing in the second current
$\Phi_{\frac{1}{2}}^{(h_{1}+h_{2}-2-h=1),i}(w)$ of (4.12) by substituting
$q_{F}^{4,\frac{5}{2},3}$ and $q_{B}^{4,\frac{5}{2},3}$ correctly. On the
other hand, there exists the current $G^{i}(w)$ term of ${\cal N}=4$ stress
energy tensor with the structure constant
$\frac{128}{3}(\lambda-1)\lambda(2\lambda-1)(2\lambda+1)$ having the $\lambda$
factor. We can check that this structure constant is equal to the one in the
first current terms in (4.12) where the previous $q_{F}^{4,\frac{5}{2},3}$ and
$q_{B}^{4,\frac{5}{2},3}$ are replaced by $q_{F}^{3,\frac{5}{2},3}$ and
$q_{B}^{3,\frac{5}{2},3}$ respectively. The weight $h_{1}$ is replaced by the
weight $(h_{1}-1)$ and this reflects the fact that the weight $h_{1}$ is
increased by $1$ from $h_{1}=h_{2}+2$ to $h_{1}=h_{2}+3$. Now we have seen the
extra structures on the right hand side of this particular pole for the
weights we are considering..

###  4.5 The commutator relation between the lowest component and the last
component with $h_{1}=h_{2}+1,h_{2}+2,h_{2}+3$ for nonzero $\lambda$

Finally, the commutator relation between the lowest component
$\Phi_{0}^{(h_{1})}(z)$ of the $h_{1}$-th ${\cal N}=4$ multiplet and the last
component (and derivative term) $\tilde{\Phi}_{2}^{(h_{2})}(w)$ of the
$h_{2}$-th ${\cal N}=4$ multiplet can be determined by using (4.1)

$\displaystyle\big{[}(\Phi^{(h_{1})}_{0})_{m},(\tilde{\Phi}^{(h_{2})}_{2})_{n}\big{]}=\frac{(-4)^{h_{1}-2}}{(2h_{1}-1)}\,4(-4)^{h_{2}-4}\Bigg{[}$
$\displaystyle\left(\begin{array}[]{c}m+h_{1}-1\\\ h_{1}+h_{2}-1\\\
\end{array}\right)\,q^{h_{1}+h_{2}-2}\,\Bigg{(}2(h_{1}-2\lambda)\,c_{F}-2(h_{1}-1+2\lambda)\,c_{B}\Bigg{)}\,\delta_{m+n}$
(4.15) $\displaystyle+\sum_{h=0,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-1}\frac{2\,q^{h}}{(-4)^{h_{1}+h_{2}-h-2}}\,\Bigg{(}-(h_{1}-2\lambda)\,p_{\mathrm{F}}^{h_{1},h_{2}+2,h}(m,n,\lambda)$
$\displaystyle-(h_{1}-1+2\lambda)\,p_{\mathrm{B}}^{h_{1},h_{2}+2,h}(m,n,\lambda)\,\Bigg{)}\,(\Phi_{0}^{(h_{1}+h_{2}-h)})_{m+n}$
$\displaystyle+\sum_{h=0,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-1}\,\frac{q^{h}}{4(2h_{1}+2h_{2}-2h-1)(-4)^{h_{1}+h_{2}-h-6}}\,\Bigg{(}$
$\displaystyle-(h_{1}-2\lambda)(h_{1}+h_{2}-h-1+2\lambda)\,p_{\mathrm{F}}^{h_{1},h_{2}+2,h}(m,n,\lambda)$
$\displaystyle+(h_{1}-1+2\lambda)(h_{1}+h_{2}-h-2\lambda)\,p_{\mathrm{B}}^{h_{1},h_{2}+2,h}(m,n,\lambda)\,\Bigg{)}\,(\tilde{\Phi}_{2}^{(h_{1}+h_{2}-h-2)})_{m+n}\Bigg{]}\,.$
(4.16)

The field contents appearing in (4.16) are the same as the one in (4.7). As
before, we observe that for the maximum value of dummy variable $h$,
$h=h_{1}+h_{2}-1$ with odd $(h_{1}+h_{2})$, on the right hand side of (4.16),
there appear the currents $\Phi_{0}^{(1)}$ and $\tilde{\Phi}_{2}^{-1}$ which
is related to the previous current $U$ of ${\cal N}=4$ stress energy tensor.
On the other hand, for even $(h_{1}+h_{2})$, the maximum value of the weight
$h$ is given by $h=h_{1}+h_{2}-2$ because the weight $h$ should be even. For
this value, there appear the currents $\Phi_{0}^{(2)}$ and
$\tilde{\Phi}_{2}^{(0)}$, which is related to the previous current $L$, with
proper $\lambda$ dependent structure constants on the right hand side of
(4.16). Note that as before we have vanishing structure constant
$p_{B}^{h_{1},h_{2}+2,h_{1}+h_{2}-1}$ at $\lambda=0$ [8]. We comment on the
other possible cases for the weights 101010 From the OPE between
$\Phi^{(h_{1}=4)}_{0}(z)$ and $\Phi^{(h_{2}=0)}_{2}(w)$ where the weights
satisfy $h_{1}=h_{2}+4$, the fifth order pole of this OPE provides the
structure constant $\frac{8192}{7}\lambda(2\lambda-1)(4\lambda-1)$ appearing
in the current $\Phi^{(1)}_{0}(w)$ and the structure constant
$\frac{256}{7}\lambda(2\lambda-1)(10\lambda^{2}-5\lambda-2)$ appearing in the
current $U(w)$. In (4.16), the exponent of the first current implies the
weight $h_{1}+h_{2}-h=4-h$. Moreover, the weight $h$ is given by $h=0$ or
$h=2$ from the summation. Therefore, the current $\Phi^{(1)}_{0}(w)$ cannot
appear from the (4.16). Furthermore, the exponent of the second current
implies the weight $h_{1}+h_{2}-2-h=2-h=-1$ from the presence of $U$.
Moreover, the weight $h$ is given by $h=0$ or $h=2$ from the summation. In
this case also, the current $U(w)$ cannot appear from the (4.16). Therefore,
there appear the extra terms for the currents having the weights we are
considering which do not satisfy the above constraints..

In Appendix $G$, the remaining (anti) commutator relations are given.
Therefore, the (anti)commutator relations between the ${\cal N}=4$ multiplets
are summarized by (4.7), (4.9), (4.11), (4.12) and (4.16) in addition to
Appendices (G.4), (G.5), (G.9), (G.11), (G.15), (G.18), (G.19), (G.23), (G.25)
and (G.29). Among these, the more fundamental (anti)commutator relations are
given by (4.7), (4.9), (4.11), Appendices (G.4), and (G.15) in the sense that
the field contents on the right hand sides of the remaining ones can be seen
from the ones of these fundamental relations up to signs. As mentioned at the
end of previous section $3$, by using (4.7), (4.16) and Appendix (G.29)
together with (4.1), the $w_{1+\infty}$ algebra can be obtained by taking the
proper limit on the parameter $q$ at $\lambda=0$ with the contractions of the
currents.

Acknowledgments

This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No. 2023R1A2C1003750).

##  Appendix A The structure constants

By introducing the generalized hypergeometric function

$\displaystyle\phi_{r}^{h_{1},h_{2}}(\Lambda,a)\equiv\
_{4}F_{3}\left[\begin{array}[]{c|}\frac{1}{2}+\Lambda\ ,\frac{1}{2}-\Lambda\
,\frac{1+a-r}{2}\ ,\frac{a-r}{2}\\\ \frac{3}{2}-h_{1}\ ,\frac{3}{2}-h_{2}\
,\frac{1}{2}+h_{1}+h_{2}-r\end{array}\ 1\right],$ (A.3)

and mode dependent function

$\displaystyle N^{h_{1},h_{2}}_{h}(m,n)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\sum_{l=0}^{h+1}(-1)^{l}\left(\begin{array}[]{c}h+1\\\ l\\\
\end{array}\right)[h_{1}-1+m]_{h+1-l}[h_{1}-1-m]_{l}$ (A.6)
$\displaystyle\times\\!$ $\displaystyle[h_{2}-1+n]_{l}[h_{2}-1-n]_{h+1-l},$
(A.7)

the three kinds of structure constants in [13] can be summarized, together
with (A.3) and (A.7), by

$\displaystyle\mathrm{BB}^{h_{1},h_{2}}_{r,\,\pm}(m,n;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1},h_{2}}(m,n)\Bigg{[}\phi_{r}^{h_{1},h_{2}}(\mu,1)\pm\phi_{r}^{h_{1},h_{2}}(1-\mu,1)\Bigg{]},$
$\displaystyle\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{r,\,\pm}(m,\rho;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1},h_{2}+\frac{1}{2}}(m,\rho)\Bigg{[}\phi_{r+1}^{h_{1},h_{2}+1}(\mu,\frac{3\pm
1}{2})$ $\displaystyle\pm$
$\displaystyle\phi_{r+1}^{h_{1},h_{2}+1}(1-\mu,\frac{3\pm 1}{2})\Bigg{]},$
$\displaystyle\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{r,\,\pm}(\rho,\omega;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(\rho,\omega)\Bigg{[}\phi_{r+1}^{h_{1}+1,h_{2}+1}(\mu,\frac{3\pm
1}{2})$ (A.8) $\displaystyle\pm$
$\displaystyle\phi_{r+1}^{h_{1}+1,h_{2}+1}(1-\mu,\frac{3\pm 1}{2})\Bigg{]}.$

Then the structure constants in section $3$, from (A.8), are given by

$\displaystyle p_{F,h}^{h_{1},h_{2}}(m,n,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\Bigg{[}\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,+}+\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle p_{B,h}^{h_{1},h_{2}}(m,n,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\Bigg{[}\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,+}-\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{F,2h}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{(2h_{1}-2h-3)}{16(h+1)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{F,2h+1}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,+}-\frac{(h_{1}-h-2)}{4(2h+3)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{B,2h}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,+}-\frac{(2h_{1}-2h-3)}{16(h+1)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{B,2h+1}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,+}-\frac{(h_{1}-h-2)}{4(2h+3)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{F,2h}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,+}-\frac{2(h_{1}+h_{2}-h)}{(2h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{F,2h+1}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{2(h_{1}+h_{2}-h)-1}{2(h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{B,2h}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,+}+\frac{2(h_{1}+h_{2}-h)}{(2h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{B,2h+1}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{2(h_{1}+h_{2}-h)-1}{(2h+2)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda}.$
(A.9)

At $\lambda=0$, the above structure constants reduce to the ones in [6, 12].

##  Appendix B The other central terms

As done in (3.11), we obtain the following central terms

$\displaystyle
V_{\lambda,\bar{a}b}^{(h_{1})+}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)\Bigg{|}_{\frac{1}{(z-w)^{h_{1}+h_{2}}}}=N\,\delta_{b\bar{c}}\,\delta_{d\bar{a}}\,\sum_{j=0}^{h_{1}-1}\,\sum_{i=0}^{h_{2}-1}\,\sum_{t=0}^{j+1}\,$
$\displaystyle\times\Bigg{(}-\frac{(h_{2}-1+2\lambda)}{(2h_{2}-1)}\,a^{j}(h_{1},\lambda)\,a^{i}(h_{2},\lambda)-\frac{(h_{2}-2\lambda)}{(2h_{2}-1)}\,a^{j}(h_{1},\lambda+\frac{1}{2})\,a^{i}(h_{2},\lambda+\frac{1}{2})\Bigg{)}$
$\displaystyle\times\frac{j!\,(t+i)!}{t!\,(t+i+1)!}\,(-1)^{h_{1}+t}\,(j+1-t)_{h_{1}-1-j}\,(h_{1}-t)_{t+1+i}\,(h_{1}+1+i)_{h_{2}-1-i}\,,$
$\displaystyle
V_{\lambda,\bar{a}b}^{(h_{1})-}(z)\,V_{\lambda,\bar{c}d}^{(h_{2})-}(w)\Bigg{|}_{\frac{1}{(z-w)^{h_{1}+h_{2}}}}=N\,\delta_{b\bar{c}}\,\delta_{d\bar{a}}\,\sum_{j=0}^{h_{1}-1}\,\sum_{i=0}^{h_{2}-1}\,\sum_{t=0}^{j+1}\,$
$\displaystyle\times\Bigg{(}\frac{(h_{1}-1+2\lambda)}{(2h_{1}-1)}\,\frac{(h_{2}-1+2\lambda)}{(2h_{2}-1)}\,a^{j}(h_{1},\lambda)\,a^{i}(h_{2},\lambda)$
$\displaystyle-\frac{(h_{1}-2\lambda)}{(2h_{1}-1)}\,\frac{(h_{2}-2\lambda)}{(2h_{2}-1)}\,a^{j}(h_{1},\lambda+\frac{1}{2})\,a^{i}(h_{2},\lambda+\frac{1}{2})\Bigg{)}$
$\displaystyle\times\frac{j!\,(t+i)!}{t!\,(t+i+1)!}\,(-1)^{h_{1}+t}\,(j+1-t)_{h_{1}-1-j}\,(h_{1}-t)_{t+1+i}\,(h_{1}+1+i)_{h_{2}-1-i}\,,$
(B.1)

where the previous relations in (2.11) are used.

##  Appendix C The OPEs between the nonsinglet currents

In this Appendix, the three kinds of OPEs corresponding to the ones in section
$3$ are given explicitly.

###  C.1  The OPE between the bosonic currents

We present the explicit check for (3.4) and (3.16) with the weights
$h_{1}=h_{2}=4$.

####  C.1.1  The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and itself

For the same indices $\hat{A}=\hat{B}=1$ in (3.4), we obtain the following
result

$\displaystyle(W_{F,4}^{\lambda,12}+W_{F,4}^{\lambda,21})({z})\,(W_{F,4}^{\lambda,12}+W_{F,4}^{\lambda,21})({w})=$
$\displaystyle\frac{1}{({z}-{w})^{8}}\,\Bigg{[}-\frac{1536}{5}(112\lambda^{6}-280\lambda^{4}+147\lambda^{2}-9)\Bigg{]}$
$\displaystyle+\frac{1}{({z}-{w})^{6}}\,\Bigg{[}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\Bigg{]}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{5}}\,\frac{1}{2}\,\Bigg{[}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\Bigg{]}\,{\partial}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{4}}\,\Bigg{[}\frac{3}{20}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\,{\partial}^{2}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})$
$\displaystyle-\frac{96}{5}(4\lambda^{2}-19)\,(W_{F,4}^{\lambda,11}+W_{F,4}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{3}}\,\Bigg{[}\frac{1}{30}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\,{\partial}^{3}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})$
$\displaystyle-\frac{1}{2}\,\frac{96}{5}(4\lambda^{2}-19)\,{\partial}\,(W_{F,4}^{\lambda,11}+W_{F,4}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{2}}\,\Bigg{[}\frac{1}{168}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\,{\partial}^{4}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})$
$\displaystyle-\frac{5}{36}\,\frac{96}{5}(4\lambda^{2}-19)\,{\partial}^{2}\,(W_{F,4}^{\lambda,11}+W_{F,4}^{\lambda,22})+6\,(W_{F,6}^{\lambda,11}+W_{F,6}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})}\,\Bigg{[}\frac{1}{1120}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+3)\,{\partial}^{5}\,(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})$
$\displaystyle-\frac{1}{36}\,\frac{96}{5}(4\lambda^{2}-19)\,{\partial}^{3}\,(W_{F,4}^{\lambda,11}+W_{F,4}^{\lambda,22})+\frac{1}{2}\,6\,{\partial}\,(W_{F,6}^{\lambda,11}+W_{F,6}^{\lambda,22})\Bigg{]}({w})+\cdots$
$\displaystyle=\frac{1}{({z}-{w})^{8}}\,\Bigg{[}-\frac{1536}{5}(112\lambda^{6}-280\lambda^{4}+147\lambda^{2}-9)\Bigg{]}$
$\displaystyle-
p_{F,4}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{F,2}^{\lambda,11}+W_{F,2}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}$
$\displaystyle-
p_{F,2}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{F,4}^{\lambda,11}+W_{F,4}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}-p_{F,0}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{F,6}^{\lambda,11}+W_{F,6}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,,$
$\displaystyle=\frac{1}{({z}-{w})^{8}}\,c_{F}(4,4,\lambda)\,\delta^{11}\,q^{4}$
$\displaystyle+\sum_{h=0,\mbox{\footnotesize
even}}^{4}\,q^{h}\,(-1)^{h-1}\,p_{F,h}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{F,6-h}^{\lambda,11}+W_{F,6-h}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,.$
(C.1)

The right hand side of (C.1) can be seen from the last two terms in (3.4) by
changing the commutator to the corresponding OPE. Note that there is an
additional factor $(-1)^{h-1}$ in the above.

####  C.1.2  The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and
$W_{F,4}^{\lambda,\hat{B}=2}$

For the different indices $\hat{A}=1$ and $\hat{B}=2$ in (3.4), the following
result is satisfied

$\displaystyle(W_{F,4}^{\lambda,12}+W_{F,4}^{\lambda,21})({z})\,i\,(W_{F,4}^{\lambda,12}-W_{F,4}^{\lambda,21})({w})=$
$\displaystyle\frac{1}{({z}-{w})^{7}}\,\Bigg{[}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)\Bigg{]}\,i\,(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{6}}\Bigg{[}\frac{1}{2}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)\Bigg{]}i\partial(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{5}}\,\Bigg{[}\frac{1}{6}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)\,i\,{\partial}^{2}\,(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})$
$\displaystyle-\frac{256}{25}\,(2\lambda-3)(2\lambda+3)(2\lambda^{2}-17)\,i\,(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{4}}\,\Bigg{[}\frac{1}{24}\,\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)\,i\,{\partial}^{3}\,(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})$
$\displaystyle-\frac{1}{2}\,\frac{256}{25}\,(2\lambda-3)(2\lambda+3)(2\lambda^{2}-17)\,i\,\partial\,(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{3}}\Bigg{[}\frac{1}{120}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)i{\partial}^{4}(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})$
$\displaystyle-\frac{1}{7}\frac{256}{25}(2\lambda-3)(2\lambda+3)(2\lambda^{2}-17)i\partial^{2}(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})+\frac{4}{5}(4\lambda^{2}-69)i(W_{F,5}^{\lambda,11}-W_{F,5}^{\lambda,22})\Bigg{]}$
$\displaystyle+\frac{1}{({z}-{w})^{2}}\,\Bigg{[}\frac{1}{720}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)i{\partial}^{5}(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})$
$\displaystyle-\frac{5}{168}\,\frac{256}{25}\,(2\lambda-3)(2\lambda+3)(2\lambda^{2}-17)\,i\,\partial^{3}\,(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})$
$\displaystyle+\frac{1}{2}\,\frac{4}{5}(4\lambda^{2}-69)\,i\,\partial\,(W_{F,5}^{\lambda,11}-W_{F,5}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})}\,\Bigg{[}\frac{1}{5040}\frac{2048}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda-1)(2\lambda+1)(2\lambda+3)i{\partial}^{6}(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})$
$\displaystyle-\frac{5}{1008}\,\frac{256}{25}\,(2\lambda-3)(2\lambda+3)(2\lambda^{2}-17)\,i\,\partial^{4}\,(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})$
$\displaystyle+\frac{3}{22}\,\frac{4}{5}\,(4\lambda^{2}-69)\,i\,\partial^{2}\,(W_{F,5}^{\lambda,11}-W_{F,5}^{\lambda,22})-\frac{1}{2}\,i\,(W_{F,7}^{\lambda,11}-W_{F,7}^{\lambda,22})\Bigg{]}({w})+\cdots$
$\displaystyle=-p_{F,5}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{F,1}^{\lambda,11}-W_{F,1}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}-p_{F,3}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{F,3}^{\lambda,11}-W_{F,3}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}$
$\displaystyle-
p_{F,1}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{F,5}^{\lambda,11}-W_{F,5}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}-p_{F,-1}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{F,7}^{\lambda,11}-W_{F,7}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,$
$\displaystyle=-\sum_{h=-1,\mbox{\footnotesize
odd}}^{5}\,q^{h}\,(-1)^{h-1}\,p_{F,h}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)i\,f^{123}\,\Bigg{[}\frac{W_{F,6-h}^{\lambda,\hat{A}=3}({w})}{({z}-{w})}\Bigg{]}+\cdots\,.$
(C.2)

Therefore, we observe that this can be seen from the first term in (3.4). It
turns out that the three elements of $W_{F,4}^{\lambda,\hat{A}}$ are
normalized correctly. Here the structure constant is given by $f^{123}=1$. We
can also check other cases of (3.4) by choosing the different indices and
expect to have similar relations to (C.1) and (C.2). We have checked that the
relation (3.4) satisfies for the case of the arbitrary weight $h_{1}$ with the
restricted weight $h_{2}$ ($h_{1}=h_{2},h_{2}\pm 1$).

####  C.1.3 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and itself

We can consider the second case described by (3.16). It turns out that we
obtain the result

$\displaystyle(W_{B,4}^{\lambda,12}+W_{B,4}^{\lambda,21})({z})\,(W_{B,4}^{\lambda,12}+W_{B,4}^{\lambda,21})({w})=$
$\displaystyle\frac{1}{({z}-{w})^{8}}\,\Bigg{[}\frac{6144}{5}(28\lambda^{6}-84\lambda^{5}+35\lambda^{4}+70\lambda^{3}-42\lambda^{2}-7\lambda+3)\Bigg{]}$
$\displaystyle+\frac{1}{({z}-{w})^{6}}\,\Bigg{[}\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1)\Bigg{]}\,(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{5}}\,\frac{1}{2}\,\Bigg{[}\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1)\Bigg{]}\,{\partial}\,(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{4}}\,\Bigg{[}\frac{3}{20}\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1){\partial}^{2}\,(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})$
$\displaystyle-\frac{192}{5}(2\lambda^{2}-2\lambda-9)(W_{B,4}^{\lambda,11}+W_{B,4}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{3}}\Bigg{[}\frac{1}{30}\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1){\partial}^{3}(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})$
$\displaystyle-\frac{1}{2}\frac{192}{5}(2\lambda^{2}-2\lambda-9){\partial}(W_{B,4}^{\lambda,11}+W_{B,4}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{2}}\,\Bigg{[}\frac{1}{168}\,\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1)\,{\partial}^{4}\,(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})$
$\displaystyle-\frac{5}{36}\,\frac{192}{5}(2\lambda^{2}-2\lambda-9)\,{\partial}^{2}\,(W_{B,4}^{\lambda,11}+W_{B,4}^{\lambda,22})+6\,(W_{B,6}^{\lambda,11}+W_{B,6}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})}\,\Bigg{[}\frac{1}{1120}\,\frac{2048}{5}(\lambda-2)(\lambda+1)(2\lambda-3)(2\lambda+1)\,{\partial}^{5}\,(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})$
$\displaystyle-\frac{1}{36}\,\frac{192}{5}(2\lambda^{2}-2\lambda-9)\,{\partial}^{3}\,(W_{B,4}^{\lambda,11}+W_{B,4}^{\lambda,22})+\frac{1}{2}\,6\,{\partial}\,(W_{B,6}^{\lambda,11}+W_{B,6}^{\lambda,22})\Bigg{]}({w})+\cdots$
$\displaystyle=\frac{1}{({z}-{w})^{8}}\,\Bigg{[}\frac{6144}{5}(28\lambda^{6}-84\lambda^{5}+35\lambda^{4}+70\lambda^{3}-42\lambda^{2}-7\lambda+3)\Bigg{]}$
$\displaystyle-
p_{B,4}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{B,2}^{\lambda,11}+W_{B,2}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}-p_{B,2}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{B,4}^{\lambda,11}+W_{B,4}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}$
$\displaystyle-
p_{B,0}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{B,6}^{\lambda,11}+W_{B,6}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,$
$\displaystyle=\frac{1}{({z}-{w})^{8}}\,c_{B}(4,4,\lambda)\,\delta^{11}\,q^{4}$
$\displaystyle+\sum_{h=0,\mbox{\footnotesize
even}}^{4}\,q^{h}\,(-1)^{h-1}\,p_{B,h}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{(W_{B,6-h}^{\lambda,11}+W_{B,6-h}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,,$
(C.3)

which looks similar to the previous one in (C.1). The corresponding central
charge, the structure constants and the currents occur.

####  C.1.4 The OPE between $W_{B,4}^{\lambda,\hat{A}=1}$ and
$W_{B,4}^{\lambda,\hat{B}=2}$

By taking the different indices $\hat{A}=1$ and $\hat{B}=2$ in (3.16), the
following result is obtained

$\displaystyle(W_{B,4}^{\lambda,12}+W_{B,4}^{\lambda,21})({z})\,i\,(W_{B,4}^{\lambda,12}-W_{B,4}^{\lambda,21})({w})=$
$\displaystyle\frac{1}{({z}-{w})^{7}}\,\Bigg{[}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)\Bigg{]}\,i\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{6}}\,\frac{1}{2}\,\Bigg{[}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)\Bigg{]}\,i\,{\partial}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{5}}\,\Bigg{[}\frac{1}{6}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)i\,{\partial}^{2}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$
$\displaystyle-\frac{512}{25}(\lambda-2)(\lambda+1)(4\lambda^{2}-4\lambda-33)\,i\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{4}}\,\Bigg{[}\frac{1}{24}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)i\,{\partial}^{3}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$
$\displaystyle-\frac{1}{2}\,\frac{512}{25}(\lambda-2)(\lambda+1)(4\lambda^{2}-4\lambda-33)\,i\,\partial\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{3}}\,\Bigg{[}\frac{1}{120}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)i\,{\partial}^{4}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$
$\displaystyle-\frac{1}{7}\,\frac{512}{25}(\lambda-2)(\lambda+1)(4\lambda^{2}-4\lambda-33)\,i\,\partial^{2}\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})$
$\displaystyle+\frac{16}{5}(\lambda^{2}-\lambda-17)\,i\,(W_{B,5}^{\lambda,11}-W_{B,5}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{2}}\,\Bigg{[}\frac{1}{720}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)i\,{\partial}^{5}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$
$\displaystyle-\frac{5}{168}\,\frac{512}{25}(\lambda-2)(\lambda+1)(4\lambda^{2}-4\lambda-33)\,i\,\partial^{3}\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})$
$\displaystyle+\frac{1}{2}\,\frac{16}{5}(\lambda^{2}-\lambda-17)\,i\,\partial\,(W_{B,5}^{\lambda,11}-W_{B,5}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})}\,\Bigg{[}\frac{1}{5040}\frac{8192}{5}(\lambda-2)(\lambda-1)\lambda(\lambda+1)(2\lambda-3)(2\lambda+1)i\,{\partial}^{6}\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$
$\displaystyle-\frac{5}{1008}\,\frac{512}{25}(\lambda-2)(\lambda+1)(4\lambda^{2}-4\lambda-33)\,i\,\partial^{4}\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})$
$\displaystyle+\frac{3}{22}\,\frac{16}{5}(\lambda^{2}-\lambda-17)\,i\,\partial^{2}\,(W_{B,5}^{\lambda,11}-W_{B,5}^{\lambda,22})-\frac{1}{2}\,i\,(W_{B,7}^{\lambda,11}-W_{B,7}^{\lambda,22})\Bigg{]}({w})+\cdots$
$\displaystyle=-p_{B,5}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}$
$\displaystyle-
p_{B,3}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{B,3}^{\lambda,11}-W_{B,3}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}-p_{B,1}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{B,5}^{\lambda,11}-W_{B,5}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}$
$\displaystyle-
p_{B,-1}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)\Bigg{[}\frac{i\,(W_{B,7}^{\lambda,11}-W_{B,7}^{\lambda,22})({w})}{({z}-{w})}\Bigg{]}+\cdots\,$
$\displaystyle=-\sum_{h=-1,\mbox{\footnotesize
odd}}^{5}\,q^{h}\,(-1)^{h-1}\,p_{B,h}^{4,4}({\partial}_{{z}},{\partial}_{{w}},\lambda)i\,f^{123}\,\Bigg{[}\frac{W_{B,6-h}^{\lambda,\hat{A}=3}({w})}{({z}-{w})}\Bigg{]}+\cdots\,.$
(C.4)

Note that there is a $\lambda$ factor in the structure constants appearing the
current $(W_{B,1}^{\lambda,11}-W_{B,1}^{\lambda,22})$ and its descendants. It
is obvious that they vanish at $\lambda=0$.

###  C.2 The OPE between the bosonic currents and the fermionic currents

We present the explicit check for (3.18) with the weights $h_{1}=4$ and
$h_{2}=3$.

####  C.2.1 The OPE between $W_{F,4}^{\lambda,\hat{A}=1}$ and
$Q_{\frac{7}{2}}^{\lambda,\hat{B}=1}$

For the same indices $\hat{A}=\hat{B}=1$ in the first equation of (3.18), the
following result holds

$\displaystyle(W_{F,4}^{\lambda,12}+W_{F,4}^{\lambda,21})({z})\,(Q_{\frac{7}{2}}^{\lambda,12}+Q_{\frac{7}{2}}^{\lambda,21})({w})=$
$\displaystyle\frac{1}{({z}-{w})^{6}}\,\Bigg{[}\frac{256}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(2\lambda+3)\Bigg{]}\,(Q_{\frac{3}{2}}^{\lambda,11}+Q_{\frac{3}{2}}^{\lambda,22})({w})$
$\displaystyle+\frac{1}{({z}-{w})^{5}}\,\Bigg{[}\frac{2}{3}\frac{256}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(2\lambda+3)\,{\partial}\,(Q_{\frac{3}{2}}^{\lambda,11}+Q_{\frac{3}{2}}^{\lambda,22})$
$\displaystyle-\frac{256}{25}(\lambda-4)(\lambda+1)(2\lambda-3)(2\lambda+3)\,(Q_{\frac{5}{2}}^{\lambda,11}+Q_{\frac{5}{2}}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{4}}\,\Bigg{[}\frac{1}{4}\frac{256}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(2\lambda+3)\,{\partial}^{2}\,(Q_{\frac{3}{2}}^{\lambda,11}+Q_{\frac{3}{2}}^{\lambda,22})$
$\displaystyle-\frac{3}{5}\,\frac{256}{25}(\lambda-4)(\lambda+1)(2\lambda-3)(2\lambda+3)\,{\partial}\,(Q_{\frac{5}{2}}^{\lambda,11}+Q_{\frac{5}{2}}^{\lambda,22})$
$\displaystyle-\frac{16}{25}(2\lambda+3)(4\lambda^{2}+18\lambda-61)\,(Q_{\frac{7}{2}}^{\lambda,11}+Q_{\frac{7}{2}}^{\lambda,22})\Bigg{]}({w})$
$\displaystyle+\frac{1}{({z}-{w})^{3}}\,\Bigg{[}\frac{1}{15}\frac{256}{5}(\lambda-1)(\lambda+1)(2\lambda-3)(2\lambda+1)(2\lambda+3)\,{\partial}^{3}\,(Q_{\frac{3}{2}}^{\lambda,11}+Q_{\frac{3}{2}}^{\lambda,22})$