A Supersymmetric Extension of $w_{1+\infty}$ Algebra

in the Celestial Holography

Changhyun Ahn† and Man Hea Kim∗

$\dagger$ Department of Physics, Kyungpook National University, Taegu 41566,
Korea

$\ast$ Department of Physics Education, Sunchon National University, Sunchon
57922, Korea

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We determine the ${\cal N}=1$ supersymmetric topological $W_{\infty}$ algebra
by using the $\lambda$ deformed bosons $(\beta,\gamma)$ and fermions $(b,c)$
ghost system. By considering the real bosons and the real fermions at
$\lambda=0$ (or $\lambda=\frac{1}{2}$), the ${\cal N}=1$ supersymmetric
$W_{\frac{\infty}{2}}$ algebra is obtained. At $\lambda=\frac{1}{4}$, other
${\cal N}=1$ supersymmetric $W_{1+\infty}[\lambda=\frac{1}{4}]$ algebra is
determined. We also obtain the extension of Lie superalgebra $PSU(2,2|{\cal
N}=4)$ appearing in the worldsheet theory by using the symplectic bosons and
the fermions. We identify the soft current algebra between the graviton, the
gravitino, the photon (the gluon), the photino (the gluino) or the scalars,
equivalent to ${\cal N}=1$ supersymmetric $W_{1+\infty}[\lambda]$ algebra, in
two dimensions with the ${\cal N}=1$ supergravity theory in four dimensions
discovered by Freedman, van Nieuwenhuizen and Ferrara in 1976 and its matter
coupled theories, via celestial holography.

###### Contents

  1. 1  Introduction
  2. 2  A $\lambda$ deformation of the ${\cal N}=1$ supersymmetric topological $W_{\infty}^{\text{top}}$ algebra
    1. 2.1 The bosonic currents
    2. 2.2 The commutator between the bosonic currents
    3. 2.3 The three commutators between the bosonic currents and the fermionic currents
    4. 2.4 The $q\rightarrow 0$ limit
  3. 3  A truncation of ${\cal N}=2$ supersymmetric $W_{1+\infty}[\lambda]$ algebra
    1. 3.1  At $\lambda=0$
      1. 3.1.1 The bosonic and fermionic currents
      2. 3.1.2 The five OPEs (and (anti)commutators)
      3. 3.1.3 The $q\rightarrow 0$ limit
    2. 3.2  At $\lambda=\frac{1}{4}$
    3. 3.3  At $\lambda=\frac{1}{2}$
    4. 3.4  At a general $\lambda$
      1. 3.4.1 The $q\rightarrow 0$ limit
  4. 4  An extension of Lie superalgebra $PSU(2,2|4)$
    1. 4.1 The extension of Lie superalgebra $PSU(2,2|4)$
    2. 4.2  The construction of bosonic current
    3. 4.3  The OPEs between the bosonic current and other generators
    4. 4.4  The $q\rightarrow 0$ limit
  5. 5  The application in the celestial conformal field theory
    1. 5.1  The soft current algebra in section $2$
    2. 5.2 The soft current algebra in subsection $3.1$
    3. 5.3  Other soft current algebra in subsection $3.4$
      1. 5.3.1  The ${\cal N}=1$ supergravity theory
      2. 5.3.2  The ${\cal N}=1$ supersymmetric Maxwell Einstein theory
      3. 5.3.3  The ${\cal N}=1$ supergravity theory coupled to the scalar multiplet
    4. 5.4  Towards the soft current algebra in section $4$
  6. 6  Conclusions and outlook
  7. A  The seven commutators (and OPEs) of section $2$
  8. B  The details of the subsection $3.1$
    1. B.1  The five (anti)commutators in the subsection $3.1$
    2. B.2  The $q\rightarrow 0$ limit at $\lambda=0$ in the subsection $3.1$
  9. C  The (anti)commutators at $\lambda=\frac{1}{4}$ in the subsection $3.2$
  10. D  The $q\rightarrow 0$ limit at general $\lambda$ in the subsection $3.4$
  11. E  The (anti)commutators in the section $4$

##  1  Introduction

A celestial holography connects the gravitational scattering amplitudes in the
asymptotically flat four dimensional spacetimes and the conformal field theory
living on the two dimensional celestial sphere [1, 2, 3, 4, 5].

In [6], a soft current algebra between the gravitons and the gluons in the
Einstein Yang-Mills theory is obtained. The wedge subalgebra of $w_{1+\infty}$
algebra [7] appears. See also the relevant work in [8] where the previous
works in [9, 10] are used. In [11], by considering the fermionic partners,
gravitinos and gluinos as well as the above bosonic ones in the ${\cal N}=1$
supersymmetric Einstein Yang-Mills theory, the operator product
expansions(OPEs) of gravitinos and gluinos with gravitons and gluons are
determined, along the lines of [9, 10]. The corresponding ${\cal N}=1$
supersymmetric soft current algebra between the above soft particles is
obtained in [12, 13] by generalizing the works of [6, 8]. Note that the
anticommutators between the fermionic operators are vanishing.

On the other hand, in [14], the ${\cal N}=1$ superconformal algebra [15, 16]
where there exists a nontrivial anticommutator for the modes of the fermionic
current as well as the usual two commutators is reproduced from the Lie
superalgebra based on the BMS symmetries [17, 18]. It is natural to ask how we
can construct the supersymmetric soft current algebra where the
anticommutators between the fermionic operators are nonvanishing.

In [19], by considering the deformation parameter $\lambda$, the
generalization of [20] is obtained. The bosonic subalgebras are given by
$W^{K}_{1+\infty}[\lambda]\times W^{K}_{1+\infty}[\lambda+\frac{1}{2}]$. The
first factor is realized by $K$ values of $(b,c)$ fermions and the second
factor is realized by $K$ values of $(\beta,\gamma)$ bosons. The above
deformation parameter $\lambda$ appears in the weights of above fields
nontrivially. By construction, there exist several anticommutators between the
modes for the fermionic currents [19].

In this paper, we would like to construct the following things as follows.

i) One ${\cal N}=1$ supersymmetric extension of $w_{1+\infty}$ algebra is
found in the work of [21]. Because the ${\cal N}=2$ supersymmetric
$W_{1+\infty}[\lambda]$ algebra for generic $\lambda$ parameter exists in
[19], the corresponding ${\cal N}=1$ supersymmetric topological
$W_{\infty}^{\text{top}}$ algebra with $\lambda$ deformation, which is a
generalization of [21], can be determined explicitly. ii) Other type of ${\cal
N}=1$ supersymmetric $W_{\frac{\infty}{2}}$ algebra is studied [22, 23]
without any explicit structure constants. By introducing the real free bosons
and fermions with $\lambda=0$ (or $\lambda=\frac{1}{2}$), half of the bosonic
currents and half of the fermionic currents can survive. The explicit
(anti)commutator relations (or OPEs) for ${\cal N}=1$ supersymmetric
$W_{\frac{\infty}{2}}$ algebra can be obtained. iii) Furthermore, at the
particular value of $\lambda=\frac{1}{4}$, the ${\cal N}=1$ supersymmetry
remains in [22, 23, 24]. It is difficult to extract the structure constants
from [22, 23]. We determine the complete structure of this ${\cal N}=1$
$W_{1+\infty}[\lambda=\frac{1}{4}]$ algebra by providing the explicit
structure constants. iv) We can examine the particular limit of [19] in order
to check whether there exist the nontrivial anticommutator relations between
the modes for the fermionic currents with a general $\lambda$ dependence in
the context of ${\cal N}=1$ supersymmetric $W_{1+\infty}[\lambda]$ algebra. v)
In [25, 26, 27], the free field worldsheet realization provides the weight
$\frac{1}{2}$ conjugate pairs of symplectic boson fields and four weight
$\frac{1}{2}$ complex fermion fields. From our experience in [19], we can
determine the extension of Lie superalgebra $PSU(2,2|4)$ by considering the
derivatives acting on the quadratic free fields. vi) The asymptotic symmetry
algebra of ${\cal N}=8$ supergravity theory in four dimensions is studied in
[28, 29]. It seems that there should appear the soft current algebra in two
dimensions. As a first step, we try to describe and interpret the algebras
obtained above by taking the particular limit of the parameter and inserting
the helicities appropriately in the context of the (conformal) supergravity
theory via celestial holography.

In section $2$, we apply the procedure of [21] to the algebra studied in [19].
In section $3$, at the particular value $\lambda=0$, by using the free field
construction in [22, 23], the new algebra is obtained. For the
$\lambda=\frac{1}{4}$, other new algebra is determined. Finally, for generic
$\lambda$, we take the particular limit for the rescaling of the modes for the
currents and obtain the various (anti)commutators. In section $4$, by using
the free field construction in [25, 26], the new algebra is obtained. In
section $5$, the corresponding dual bulk theories are described in the
celestial holography. In section $6$, we summarize what we have obtained in
this paper and present some open problems. In Appendices, some details
appearing in the previous sections are described.

The Thielemans package [30] is used together with a mathematica [31] all the
times in this paper. We list some relevant works [32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] on the supersymmetric
construction in the context of [1].

##  2  A $\lambda$ deformation of the ${\cal N}=1$ supersymmetric topological
$W_{\infty}^{\text{top}}$ algebra

The operator product expansions of the $(\beta,\gamma)$ and $(b,c)$ systems
are given by [53]

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

The $(\beta,\gamma)$ fields are bosonic operators while the $(b,c)$ fields are
fermionic operators. The conformal weights of $(\beta,\gamma)$ fields are
given by $(\lambda,1-\lambda)$ while those of $(b,c)$ fields are given by
$(\frac{1}{2}+\lambda,\frac{1}{2}-\lambda)$. There are fundamental indices
$a,b$ and antifundamental indices $\bar{a},\bar{b}$ of $SU(K)$ and there are
fundamental indices $i,j$ and antifundamental indices $\bar{i},\bar{j}$ of
$SU(N)$.

###  2.1 The bosonic currents

From Appendix $(A.6)$ or $(D.1)$ of [19], we can calculate the modes for the
bosonic currents in the $\lambda$ deformed ${\cal N}=1$ supersymmetric
topological $W_{\infty}^{\text{top}}$ algebra by using the anticommutator
$(6)$ of [21] as follows:

$\displaystyle\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{(Q^{\lambda}_{h+\frac{1}{2}})_{m+\frac{1}{2}}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=1}^{h+1\,\,}\sum_{k=0}^{h-h_{3}+1}\,2(-1)^{h_{3}+1}\,(4q)^{h-h_{3}+1}\,(2h_{3}-1)!$
(2.2) $\displaystyle\times$
$\displaystyle\bigg{[}\,\,U_{F}^{\,\,h,1,h_{3},k}(\lambda)\,[m+h]_{h-h_{3}-k+1}\,[0]_{k}\,(W^{\lambda}_{F,\,h_{3}})_{m}$
$\displaystyle+$ $\displaystyle
U_{B}^{\,\,h,1,h_{3},k}(\lambda)\,[m+h]_{k}\,[0]_{h-h_{3}-k+1}\,(W^{\lambda}_{B,\,h_{3}})_{m}\bigg{]}\,$
$\displaystyle=$ $\displaystyle
2(\tilde{W}^{\lambda}_{F,\,h+1})_{m}+2(\tilde{W}^{\lambda}_{B,\,h+1})_{m}\equiv
2(\tilde{W}^{\lambda}_{\,h+1})_{m}\,,$

where the $\lambda$ deformed modes on the right hand side of (2.2) are

$\displaystyle(\tilde{W}_{F,\,h}^{\lambda})_{m}$ $\displaystyle=$
$\displaystyle(W_{F,\,h}^{\lambda})_{m}-q\frac{2(h-1)-4\lambda}{2(h-2)+1}(m+h-1)(W_{F,\,h-1}^{\lambda})_{m}\,,$
$\displaystyle(\tilde{W}_{B,\,h}^{\lambda})_{m}$ $\displaystyle=$
$\displaystyle(W_{B,\,h}^{\lambda})_{m}+q\frac{2(h-2)+4\lambda}{2(h-2)+1}(m+h-1)(W_{B,\,h-1}^{\lambda})_{m}\,.$
(2.3)

It is obvious that these lead to the previous results in [13, 21] for
$\lambda=0$. The $\lambda$ dependent terms of the coefficients on the right
hand side in (2.2) (that is, the second terms in (2.3)) come from the case of
$h_{3}=h$ while the $\lambda$ independent terms of those (the first terms in
(2.3)) appear when $h_{3}=h+1$. For the $h_{3}<h$, then there are no
contributions in the summation of (2.2). There is no central term in the above
anticommutator due to the particular combinations of the modes on the left
hand side. The falling Pochhammer symbol $[a]_{n}\equiv a(a-1)\cdots(a-n+1)$
is used. The structure constants in (2.2) are given by Appendix $C$.

In terms of currents, by using the standard procedure in the conformal field
theory, we obtain the following results corresponding to (2.3) 111By
construction, the weight $h$ is greater than or equal to $2$.

$\displaystyle\tilde{W}_{F,\,h}^{\lambda}$ $\displaystyle=$ $\displaystyle
W_{F,\,h}^{\lambda}+q\frac{2(h-1)-4\lambda}{2(h-2)+1}\partial
W_{F,\,h-1}^{\lambda}\,,\ $ $\displaystyle\tilde{W}_{B,\,h}^{\lambda}$
$\displaystyle=$ $\displaystyle
W_{B,\,h}^{\lambda}-q\frac{2(h-2)+4\lambda}{2(h-2)+1}\partial
W_{B,\,h-1}^{\lambda}.$ (2.4)

Of course, when we take $\lambda\rightarrow 0$ limit, the results in (2.4)
become those in [13, 21]. In principle, these currents can be obtained from
the OPE between the fermionic currents corresponding to the left hand side of
(2.2).

In terms of free fields appearing in (2.1), the above currents in (2.4) take
the form

$\displaystyle\tilde{W}_{F,\,h}^{\lambda}(z)$ $\displaystyle=$
$\displaystyle(-4q)^{h-2}\sum_{i=0}^{h-1}\Big{[}2a^{i}\big{(}h,\lambda+\tfrac{1}{2}\big{)}-\alpha^{i}(h,\lambda)\Big{]}\,\partial_{z}^{h-i-1}\Big{(}(\partial^{i}b^{\bar{l}b})\,\delta_{b\bar{a}}\delta_{\bar{l}l}\,c^{l\bar{a}}\Big{)}(z),$
$\displaystyle\tilde{W}_{B,\,h}^{\lambda}(z)$ $\displaystyle=$
$\displaystyle(-4q)^{h-2}\sum_{i=0}^{h-1}\Big{[}2a^{i}(h,\lambda)-\alpha^{i}(h,\lambda)\Big{]}\,\partial_{z}^{h-i-1}\Big{(}(\partial^{i}\beta^{\bar{l}b})\,\delta_{b\bar{a}}\delta_{\bar{l}l}\,\gamma^{l\bar{a}}\Big{)}(z),$
$\displaystyle\tilde{W}^{\lambda}_{h}(z)$ $\displaystyle\equiv$
$\displaystyle\tilde{W}^{\lambda}_{F,\,h}(z)+\tilde{W}^{\lambda}_{B,\,h}(z),$
$\displaystyle\tilde{W}_{F,\,h}^{\lambda,\hat{A}}(z)$ $\displaystyle=$
$\displaystyle(-4q)^{h-2}\sum_{i=0}^{h-1}\Big{[}2a^{i}\big{(}h,\lambda+\tfrac{1}{2}\big{)}-\alpha^{i}(h,\lambda)\Big{]}\,\partial_{z}^{h-i-1}\Big{(}(\partial^{i}b^{\bar{l}b})\,t^{\hat{A}}_{b\bar{a}}\delta_{\bar{l}l}\,c^{l\bar{a}}\Big{)}(z),$
$\displaystyle\tilde{W}_{B,\,h}^{\lambda,\hat{A}}(z)$ $\displaystyle=$
$\displaystyle(-4q)^{h-2}\sum_{i=0}^{h-1}\Big{[}2a^{i}(h,\lambda)-\alpha^{i}(h,\lambda)\Big{]}\,\partial_{z}^{h-i-1}\Big{(}(\partial^{i}\beta^{\bar{l}b})\,t^{\hat{A}}_{b\bar{a}}\delta_{\bar{l}l}\,\gamma^{l\bar{a}}\Big{)}(z),$
$\displaystyle\tilde{W}^{\lambda,\hat{A}}_{h}(z)$ $\displaystyle\equiv$
$\displaystyle\tilde{W}^{\lambda,\hat{A}}_{F,\,h}(z)+\tilde{W}^{\lambda,\hat{A}}_{B,\,h}(z),$
(2.5)

as well as the fermionic currents in [19]

$\displaystyle Q_{h+\frac{1}{2}}^{\lambda}(z)$ $\displaystyle=$
$\displaystyle\sqrt{2}\,(-4q)^{h-1}\sum_{i=0}^{h-1}\beta^{i}(h+1,\lambda)\,\partial_{z}^{h-i-1}((\partial_{z}^{i}b^{\bar{l}b})\,\delta_{b\bar{a}}\delta_{\bar{l}l}\,\gamma^{l\bar{a}})(z),$
$\displaystyle Q_{h+\frac{1}{2}}^{\lambda,\hat{A}}(z)$ $\displaystyle=$
$\displaystyle\sqrt{2}\,(-4q)^{h-1}\sum_{i=0}^{h-1}\beta^{i}(h+1,\lambda)\,\partial_{z}^{h-i-1}((\partial_{z}^{i}b^{\bar{l}b})\,(t^{\hat{A}})_{b\bar{a}}\delta_{\bar{l}l}\,\gamma^{l\bar{a}})(z).$
(2.6)

Note that the $\lambda$ dependent coefficient terms in (2.5) appear
nontrivially and can be expressed as the known coefficients. The various
coefficients $a^{i}(h,\lambda),\alpha^{i}(h,\lambda)$ and
$\beta^{i}(h,\lambda)$ appearing in (2.5) and (2.6) are given in [22, 23, 19]:

$\displaystyle a^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-1\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-1-i}}{(h+i)_{h-1-i}}\,,\qquad
0\leq i\leq h-1\,,$ (2.9)
$\displaystyle\alpha^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-1\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-1-i}}{(h+i-1)_{h-1-i}}\,,\qquad
0\leq i\leq h-1\,,$ (2.12)
$\displaystyle\beta^{i}(h,\lambda)\equiv\left(\begin{array}[]{c}h-2\\\ i\\\
\end{array}\right)\,\frac{(-2\lambda-h+2)_{h-2-i}}{(h+i)_{h-2-i}}\,,\qquad
0\leq i\leq h-2\,,$ (2.15)

where the rising Pochhammer symbols are used in $(a)_{n}\equiv
a(a+1)\cdots(a+n-1)$ with a nonnegative integer $n$. The parentheses in (2.15)
are the binomial coefficients. The newly obtained bosonic currents are given
by $\tilde{W}^{\lambda}_{h}$ with (2.4) and $\tilde{W}^{\lambda,\hat{A}}_{h}$
with (2.5) where $(t^{\hat{A}})_{b\bar{a}}$ is an adjoint representation for
$SU(K)$ generator. The fermionic currents $Q_{h+\frac{1}{2}}^{\lambda}$ and
$Q_{h+\frac{1}{2}}^{\lambda,\hat{A}}$ remain the same as the ones in [19].

We would like to construct the new algebra by the bosonic currents
$\tilde{W}^{\lambda}_{h}$ and $\tilde{W}^{\lambda,\hat{A}}_{h}$ and the
fermionic currents $Q_{h+\frac{1}{2}}^{\lambda}$ and
$Q_{h+\frac{1}{2}}^{\lambda,\hat{A}}$ together with (2.5) and (2.6). Because
the relative coefficients in (2.4) contain the $\lambda$ explicitly, the
structure constants in this new algebra will be different from those in [19].

###  2.2 The commutator between the bosonic currents

First of all, we should calculate the OPE between the bosonic currents
obtained in previous subsection. One way to determine that OPE between the
singlets of $SU(K)$ is as follows. By calculating $(7)$ of [21] or (2.2),

$\displaystyle
4\commutator{(\tilde{W}^{\lambda}_{\,h_{1}+1})_{m}}{(\tilde{W}^{\lambda}_{\,h_{2}+1})_{n}}$
$\displaystyle=$
$\displaystyle\commutator{\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{(Q^{\lambda}_{h_{1}+\frac{1}{2}})_{m+\frac{1}{2}}}}{\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{(Q^{\lambda}_{h_{2}+\frac{1}{2}})_{n+\frac{1}{2}}}}$
(2.16) $\displaystyle=$
$\displaystyle\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{\commutator{(Q^{\lambda}_{h_{2}+\frac{1}{2}})_{n+\frac{1}{2}}}{\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{(Q^{\lambda}_{h_{1}+\frac{1}{2}})_{m+\frac{1}{2}}}}}$
$\displaystyle=$ $\displaystyle
2\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{\commutator{(Q^{\lambda}_{h_{2}+\frac{1}{2}})_{n+\frac{1}{2}}}{(\tilde{W}^{\lambda}_{\,h_{1}+1})_{m}}},$

where the Jacobi identities are used several times and the previous relation
(2.2) is used.

Now the commutator inside of the curly bracket in (2.16) can be determined
from (2.5) and (2.6) according to (2.1) and it turns out that

$\displaystyle-2\sum_{h_{3}=1}^{h_{1}+h_{2}\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-1}\,(2h_{3})!\bigg{(}\,\,\tilde{T}_{R}^{\,\,h_{1}+1,h_{2},h_{3},k}\,\,[m+h_{1}]_{h_{1}+h_{2}-h_{3}-k}\,[n+h_{2}]_{k}$
$\displaystyle-\tilde{T}_{L}^{\,\,h_{1}+1,h_{2},h_{3},k}\,\,[m+h_{1}]_{k}\,[n+h_{2}]_{h_{1}+h_{2}-h_{3}-k}\bigg{)}\,\anticommutator{(\bar{Q}^{\lambda}_{\frac{3}{2}})_{-\frac{1}{2}}}{(Q_{h_{3}+\frac{1}{2}}^{\lambda})_{m+n+\frac{1}{2}}}.$
(2.17)

The structure constants are given in Appendix $A$ explicitly. Furthermore, the
above anticommutator can be replaced by the bosonic current from (2.2) and it
is given by

$\displaystyle-4\sum_{h_{3}=1}^{h_{1}+h_{2}\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-1}\,(2h_{3})!\bigg{(}\,\,\tilde{T}_{R}^{\,\,h_{1}+1,h_{2},h_{3},k}\,\,[m+h_{1}]_{h_{1}+h_{2}-h_{3}-k}\,[n+h_{2}]_{k}$
$\displaystyle-\tilde{T}_{L}^{\,\,h_{1}+1,h_{2},h_{3},k}\,\,[m+h_{1}]_{k}\,[n+h_{2}]_{h_{1}+h_{2}-h_{3}-k}\bigg{)}\,(\tilde{W}^{\lambda}_{\,h_{3}+1})_{m+n}.$
(2.18)

This implies that the structure constants in the OPE between the bosonic
currents are written in terms of those in the OPE between the bosonic current
and the fermionic current.

On the other hand, we obtain the commutator between the bosonic currents
directly from (2.5) with the help of (2.1)

$\displaystyle
4\commutator{(\tilde{W}^{\lambda}_{\,h_{1}+1})_{m}}{(\tilde{W}^{\lambda}_{\,h_{2}+1})_{n}}$
$\displaystyle=$ $\displaystyle
4\sum_{h_{3}=2}^{h_{1}+h_{2}+1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}+1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}}\,(2h_{3}-2)!$
(2.19) $\displaystyle\times$
$\displaystyle\bigg{(}\,\,\tilde{S}_{R}^{\,\,h_{1}+1,h_{2}+1,h_{3},k}\,\,[m+h_{1}]_{h_{1}+h_{2}-h_{3}-k+1}\,[n+h_{2}]_{k}$
$\displaystyle-$
$\displaystyle\tilde{S}_{L}^{h_{1}+1,h_{2}+1,h_{3},k}[m+h_{1}]_{k}[n+h_{2}]_{h_{1}+h_{2}-h_{3}-k+1}\bigg{)}(\tilde{W}^{\lambda}_{h_{3}})_{m+n}.$

Again, the structure constants are given in Appendix $A$.

By comparing (2.18) with (2.19), there are nontrivial relations between the
structure constants as follows

$\displaystyle\tilde{S}_{R}^{\,\,h_{1}+1,h_{2}+1,h_{3},k}(\lambda)=-\tilde{T}_{R}^{\,\,h_{1}+1,h_{2},h_{3}-1,k}(\lambda)\,,\quad\tilde{S}_{L}^{\,\,h_{1}+1,h_{2}+1,h_{3},k}(\lambda)=-\tilde{T}_{L}^{\,\,h_{1}+1,h_{2},h_{3}-1,k}(\lambda).$
(2.20)

This can be checked by considering these structure constants from Appendix $A$
directly. Then we are left with two independent structure constants in our new
algebra.

Therefore, the commutator relation between the bosonic currents is summarized
by (2.19) or (2.18). The other two commutators between the bosonic currents
having $SU(K)$ indices can be obtained similarly and they are given in
Appendix $A$ (the second and the third relations in Appendix (A.1)).

###  2.3 The three commutators between the bosonic currents and the fermionic
currents

We have seen the commutators between the bosonic singlet current and the
fermionic singlet current in (2.17). Then the remaining three commutators we
are left with can be determined by using the proper delta symbols, $f$ symbols
and $d$ symbols for the $SU(K)$ (the last three relations of (A.1)). The
complete seven commutator relations are summarized in Appendix $A$.

###  2.4 The $q\rightarrow 0$ limit

By taking the following transformations for the currents,

$\displaystyle\tilde{W}^{\lambda}_{h}$ $\displaystyle\longrightarrow$
$\displaystyle
q^{h-2}\,\tilde{W}^{\lambda}_{h}\,,\qquad\tilde{W}^{\lambda,\hat{A}}_{h}\longrightarrow
q^{h}\,\tilde{W}^{\lambda,\hat{A}}_{h}\,,\qquad$ $\displaystyle
Q^{\lambda}_{h+\frac{1}{2}}$ $\displaystyle\longrightarrow$ $\displaystyle
q^{h-\frac{3}{2}}Q^{\lambda}_{h+\frac{1}{2}}\,,\qquad
Q^{\lambda,\hat{A}}_{h+\frac{1}{2}}\longrightarrow
q^{h+\frac{1}{2}}Q^{\lambda,\hat{A}}_{h+\frac{1}{2}},$ (2.21)

we obtain the $\lambda$ deformed ${\cal N}=1$ supersymmetric $w^{K}_{\infty}$
algebra with $U(K)$ symmetry after $q\rightarrow 0$ limit in Appendix (A.1)

$\displaystyle\commutator{(\tilde{W}^{\lambda}_{h_{1}})_{m}}{(\tilde{W}^{\lambda}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(\tilde{W}^{\lambda}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda}_{h_{1}})_{m}}{(\tilde{W}^{\lambda,\hat{A}}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(\tilde{W}^{\lambda,\hat{A}}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A}}_{h_{1}})_{m}}{(\tilde{W}^{\lambda,\hat{B}}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle-\frac{i}{4}f^{\hat{A}\hat{B}\hat{C}}(\tilde{W}^{\lambda,\hat{C}}_{h_{1}+h_{2}-1})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda}_{h_{1}})_{m}}{(Q^{\lambda}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda}_{h_{1}})_{m}}{(Q^{\lambda,\hat{A}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda,\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A}}_{h_{1}})_{m}}{(Q^{\lambda}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda,\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A}}_{h_{1}})_{m}}{(Q^{\lambda,\hat{B}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle-\frac{i}{4}f^{\hat{A}\hat{B}\hat{C}}(Q^{\lambda,\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r},$
(2.22)

where $h_{1},h_{2}=2,3,\cdots$ (the weights for the fermionic currents hold
for $h_{1},h_{2}=1$), the adjoint indices
$\hat{A},\hat{B}=1,2,\cdots,(K^{2}-1)$, and the modes are $m=0,\pm 1,\pm
2,\cdots$, and $r=\pm\frac{1}{2},\pm\frac{3}{2},\cdots$. This implies that the
$\lambda$ deformation of the ${\cal N}=1$ topological
$W_{\infty}^{\text{top}}$ algebra at $q\rightarrow 0$ limit shares with common
behavior for the case without any deformation in [13] because the mode
dependent terms or constant coefficients on the right hand sides of (2.22) do
not depend on the $\lambda$. Of course, the currents themselves do depend on
the $\lambda$ explicitly. During the twisting procedure, the original weights
for the fermionic currents is increased by $\frac{1}{2}$
($h_{2}+\frac{1}{2}\rightarrow h_{2}+1$) from the fourth and fifth relations
of (2.22). The first three relations of (2.22) are the $w^{K}_{\infty}$
algebra with $U(K)$ symmetry 222As described in [13], the weight $1$ current
can be added and the corresponding $w_{1+\infty}^{K}$ algebra can be obtained
as in [54]. and the last four relations are the additional commutator
relations in its ${\cal N}=1$ supersymmetric extension 333 We can calculate
the commutators from the corresponding OPEs. For example, from the following
OPE,
$\displaystyle\tilde{W}^{\lambda}_{h_{1}}(z)\,Q^{\lambda}_{h_{2}+\frac{1}{2}}(w)$
$\displaystyle=\frac{1}{(z-w)^{2}}\,(h_{1}+h_{2}-1)\,Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}}(w)+\frac{1}{(z-w)}\,(h_{1}-1)\,\partial
Q^{\lambda}_{h_{1}+h_{2}-\frac{3}{2}}(w)+\cdots,$ we can calculate the fourth
commutator of (2.22). For $h_{1}=2$, the weight is given by $(h_{2}+1)$, not
$(h_{2}+\frac{1}{2})$..

##  3  A truncation of ${\cal N}=2$ supersymmetric $W_{1+\infty}[\lambda]$
algebra

We present the various supersymmetric algebras and their particular limits.

###  3.1  At $\lambda=0$

In this case, the conformal weights for $(b,c)$ are equal to
$(\frac{1}{2},\frac{1}{2})$ and they for $(\beta,\gamma)$ are equal to $(0,1)$
from the previous section. Then we identify $b$ with $c$ and denote it by
$\psi$ while we denote the $(\beta,\gamma)$ by $(\phi,\partial\phi)$
respectively [22, 23]. In this subsection, we use the following OPEs

$\displaystyle\psi^{i,a}(z)\,\psi^{j,b}(w)=\frac{1}{(z-w)}\delta^{ij}\delta^{ab}+\cdots\,,\quad\partial\phi^{i,a}(z)\,\partial\phi^{j,b}(w)=\frac{1}{(z-w)^{2}}\delta^{ij}\delta^{ab}+\cdots\,.$
(3.1)

There exist two independent free fields compared to the ones in previous
section. There are vector indices $a,b$ of $SO(K)$ and there are also vector
indices $i,j$ of $SO(N)$ [55] (See also [56] for the construction of real
independent free fermions.) 444The corresponding bosonic subalgebra is denoted
by $W_{\frac{1+\infty}{2}}\times W_{\frac{\infty}{2}}$ algebra where the first
factor is realized by the fermions while the second factor is realized by the
bosons in [57, 22, 23]. In the abstract, we denote its ${\cal N}=1$ version by
${\cal N}=1$ supersymmetric $W_{\frac{\infty}{2}}$ algebra. See also related
work in [58] and [59, 60] for the field contents and free field realization..

####  3.1.1 The bosonic and fermionic currents

Based on the construction of [22, 23], we determine the following currents

$\displaystyle V^{(h)+}_{ab}$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{h-1}\tilde{a}^{r}(h,0)\,(\partial^{r}\phi^{i,b})\delta_{ij}(\partial^{h-r}\phi^{j,a})+\sum_{r=0}^{h-1}\tilde{a}^{r}(h,\tfrac{1}{2})\,(\partial^{r}\psi^{i,b})\delta_{ij}(\partial^{h-1-r}\psi^{j,a})$
$\displaystyle=$
$\displaystyle\sum_{r=0}^{h-1}a^{r}(h,0)\,\partial^{h-1-r}\Big{(}(\partial^{r}\phi^{i,b})\delta_{ij}\partial\phi^{j,a}\Big{)}+\sum_{r=0}^{h-1}a^{r}(h,\tfrac{1}{2})\,\partial^{h-1-r}\Big{(}(\partial^{r}\psi^{i,b})\delta_{ij}\psi^{j,a}\Big{)},$
$\displaystyle V^{(h)-}_{ab}$ $\displaystyle=$
$\displaystyle-\frac{(h-1)}{(2h-1)}\sum_{r=0}^{h-1}\tilde{a}^{r}(h,0)\,(\partial^{r}\phi^{i,b})\delta_{ij}(\partial^{h-r}\phi^{j,a})$
$\displaystyle+$
$\displaystyle\frac{h}{(2h-1)}\sum_{r=0}^{h-1}\tilde{a}^{r}(h,\tfrac{1}{2})\,(\partial^{r}\psi^{i,b})\delta_{ij}(\partial^{h-1-r}\psi^{j,a})$
$\displaystyle=$
$\displaystyle-\frac{h-1}{2h-1}\sum_{r=0}^{h-1}a^{r}(h,0)\,\partial^{h-1-r}\Big{(}(\partial^{r}\phi^{i,b})\delta_{ij}\partial\phi^{j,a}\Big{)}$
$\displaystyle+$
$\displaystyle\frac{h}{2h-1}\sum_{r=0}^{h-1}a^{r}(h,\tfrac{1}{2})\,\partial^{h-1-r}\Big{(}(\partial^{r}\psi^{i,b})\delta_{ij}\psi^{j,a}\Big{)},$
$\displaystyle Q^{(h)\pm}_{ab}$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{h-1}\tilde{\alpha}^{r}(h,0)\,(\partial^{r}\phi^{i,b})\delta_{ij}(\partial^{h-1-r}\psi^{j,a})\mp\sum_{r=0}^{h-2}\tilde{\beta}^{r}(h,0)\,(\partial^{r}\psi^{i,b})\delta_{ij}(\partial^{h-1-r}\phi^{j,a})$
(3.2) $\displaystyle=$
$\displaystyle\sum_{r=0}^{h-1}\alpha^{r}(h,0)\,\partial^{h-1-r}\Big{(}(\partial^{r}\phi^{i,b})\delta_{ij}\psi^{j,a}\Big{)}\mp\sum_{r=0}^{h-2}\beta^{r}(h,0)\,\partial^{h-2-r}\Big{(}(\partial^{r}\psi^{i,b})\delta_{ij}\partial\phi^{j,a}\Big{)},$

where the structure constants appearing in (3.2) are given by

$\displaystyle\tilde{a}^{r}(h,\lambda)$ $\displaystyle=$
$\displaystyle\binom{h-1}{r}\,\frac{(-1)^{r}}{(h)_{h-1}}\,(2\lambda-h)_{r}\,(2-h-2\lambda)_{h-1-r}\,,$
$\displaystyle\tilde{\alpha}^{r}(h,\lambda)$ $\displaystyle=$ $\displaystyle
2\,\binom{h-1}{r}\,\frac{(-1)^{r}}{(h)_{h-1}}\,(2\lambda-h+1)_{r}\,(2-h-2\lambda)_{h-1-r}\,,$
$\displaystyle\tilde{\beta}^{r}(h,\lambda)$ $\displaystyle=$
$\displaystyle\binom{h-2}{r}\,\frac{(-1)^{r}}{(h)_{h-2}}\,(2\lambda-h+1)_{r}\,(2-h-2\lambda)_{h-2-r}\,.$
(3.3)

which appear in [22, 23]. The $h$ in the fermionic current is greater than $1$
in (3.2).

By considering the linear combinations, we obtain the following currents

$\displaystyle W_{F,h}^{ab}$ $\displaystyle=$ $\displaystyle
n_{W_{F,h}}\,\frac{(-1)^{h}}{\sum_{i=0}^{h-1}a^{i}(h,\frac{1}{2})}\Bigg{[}\frac{(h-1)}{(2h-1)}V^{(h)+}_{ab}+V^{(h)-}_{ab}\Bigg{]}\,,$
$\displaystyle W_{B,h}^{ab}$ $\displaystyle=$ $\displaystyle
n_{W_{B,h}}\,\frac{(-1)^{h}}{\sum_{i=0}^{h-1}a^{i}(h,0)}\Bigg{[}\frac{h}{2h-1}V^{(h)+}_{ab}-V^{(h)-}_{ab}\Bigg{]}\,,$
$\displaystyle Q_{h+\frac{1}{2}}^{ab}$ $\displaystyle=$ $\displaystyle
n_{Q_{h}}\,\frac{(-1)^{h+1}h}{2\sum_{i=0}^{h-1}\beta^{i}(h+1,0)}\Bigg{[}Q^{(h+1)-}_{ab}-Q^{(h+1)+}_{ab}\Bigg{]}\,,$
$\displaystyle\bar{Q}_{h+\frac{1}{2}}^{ba}$ $\displaystyle=$ $\displaystyle
n_{Q_{h}}\,\frac{(-1)^{h+1}}{2\sum_{i=0}^{h}\alpha^{i}(h+1,0)}\Bigg{[}Q^{(h+1)-}_{ab}+Q^{(h+1)+}_{ab}\Bigg{]}\,.$
(3.4)

Note that we can express these currents in terms of previous parameters in
(3.3).

By realizing the identity between the coefficients in the fermionic currents,
there exists only one independent fermionic current 555 There exists a
relation  $\displaystyle Q_{h+\frac{1}{2}}^{\,\,ab}$
$\displaystyle=(-1)^{h+1}\,\bar{Q}_{h+\frac{1}{2}}^{\,\,ab}$ for $h>1$. We
have the normalization factors  $\displaystyle
n_{W_{F},h}=q^{h-2}\,\frac{2^{h-3}(h-1)!}{(2h-3)!!}\,,\quad
n_{W_{B},h}=q^{h-2}\,\frac{2^{h-3}\,h!}{(2h-3)!!}\,,\quad
n_{Q,h}=q^{h-1}\,\frac{2^{h-\frac{1}{2}}\,h!}{(2h-1)!!}\,.$ .

####  3.1.2 The five OPEs (and (anti)commutators)

It is straightforward to calculate the following OPEs from (3.1)

$\displaystyle W^{ab}_{F,h_{1}}(z)\,W^{cd}_{F,h_{2}}(w)$ $\displaystyle=$
$\displaystyle
N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}+1}\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc}\Big{)}\,q^{2(h_{1}-2)}c_{W_{F,h_{1}}}\,\partial_{z}^{2h_{1}-1}\bigg{[}\frac{1}{(z-w)}\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,p_{F}^{h_{1},h_{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\,\bigg{[}(-1)^{h_{1}+h}\,\frac{\delta^{ac}\,W^{bd}_{F,h_{1}+h_{2}-h-2}(w)}{(z-w)}$
$\displaystyle+$
$\displaystyle\frac{\delta^{ad}\,W^{cb}_{F,h_{1}+h_{2}-h-2}(w)}{(z-w)}+(-1)^{h}\,\frac{\delta^{bc}\,W^{ad}_{F,h_{1}+h_{2}-h-2}(w)}{(z-w)}$
$\displaystyle+$
$\displaystyle(-1)^{h_{2}+h}\,\frac{\delta^{bd}\,W^{ac}_{F,h_{1}+h_{2}-h-2}(w)}{(z-w)}\bigg{]}+\cdots,$
$\displaystyle W^{ab}_{B,h_{1}}(z)\,W^{cd}_{B,h_{2}}(w)$ $\displaystyle=$
$\displaystyle
N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}+1}\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc}\Big{)}\,q^{2(h_{1}-2)}c_{W_{B,h_{1}}}\,\partial_{z}^{2h_{1}-1}\bigg{[}\frac{1}{(z-w)}\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{h=-1}^{h_{1}+h_{2}-4}q^{h}\,p_{B}^{h_{1},h_{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\bigg{[}(-1)^{h_{1}+h}\,\frac{\delta^{ac}\,W^{bd}_{B,h_{1}+h_{2}-h-2}(w)}{(z-w)}$
$\displaystyle+$
$\displaystyle\frac{\delta^{ad}\,W^{cb}_{B,h_{1}+h_{2}-h-2}(w)}{(z-w)}+(-1)^{h}\,\frac{\delta^{bc}\,W^{ad}_{B,h_{1}+h_{2}-h-2}(w)}{(z-w)}$
$\displaystyle+$
$\displaystyle(-1)^{h_{2}+h}\,\frac{\delta^{bd}\,W^{ac}_{B,h_{1}+h_{2}-h-2}(w)}{(z-w)}\bigg{]}+\cdots,$
$\displaystyle W^{ab}_{F,h_{1}}(z)\,Q^{cd}_{h_{2}+\frac{1}{2}}(w)$
$\displaystyle=$
$\displaystyle-\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,q_{F}^{h_{1},h_{2}+\frac{1}{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\,$
$\displaystyle\times$
$\displaystyle\bigg{[}(-1)^{h}\,\frac{\delta^{ad}\,Q^{cb}_{h_{1}+h_{2}-h-\frac{3}{2}}(w)}{(z-w)}+(-1)^{h_{1}+h}\,\frac{\delta^{bd}\,Q^{ca}_{h_{1}+h_{2}-h-\frac{3}{2}}(w)}{(z-w)}\bigg{]}+\cdots,$
$\displaystyle W^{ab}_{B,h_{1}}(z)\,Q^{cd}_{h_{2}+\frac{1}{2}}(w)$
$\displaystyle=$
$\displaystyle-\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,q_{B}^{h_{1},h_{2}+\frac{1}{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\,$
$\displaystyle\times$
$\displaystyle\bigg{[}(-1)^{h_{1}+h}\,\frac{\delta^{ac}\,Q^{bd}_{h_{1}+h_{2}-h-\frac{3}{2}}(w)}{(z-w)}+(-1)^{h}\,\frac{\delta^{bc}\,Q^{ad}_{h_{1}+h_{2}-h-\frac{3}{2}}(w)}{(z-w)}\bigg{]}+\cdots,$
$\displaystyle Q^{ab}_{F,h_{1}+\frac{1}{2}}(z)\,Q^{cd}_{h_{2}+\frac{1}{2}}(w)$
$\displaystyle=$ $\displaystyle
N\,\delta^{h_{1}h_{2}}\,\delta^{ac}\delta^{bd}\,\,q^{2(h_{1}-1)}\,(-1)^{h_{1}+1}\,c_{Q_{h_{1}}}\,\partial_{z}^{2h_{1}}\bigg{[}\frac{1}{(z-w)}\bigg{]}$
$\displaystyle-$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-1}q^{h}\,(-1)^{h_{1}}\,o_{F}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\,\frac{\delta^{ac}\,W^{bd}_{F,h_{1}+h_{2}-h}(w)}{(z-w)}$
$\displaystyle-$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-2}q^{h}\,(-1)^{h_{2}+h}\,o_{B}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(\partial_{z}^{m},\partial_{w}^{n},0)\,\frac{\delta^{bd}\,W^{ac}_{B,h_{1}+h_{2}-h}(w)}{(z-w)}+\cdots.$

The various structure constants are given in Appendix (B.10) and the
corresponding differential operators can be obtained by replacing the
arguments $(m,n)$ with the $m$ derivatives of $z$ and the $n$ derivatives of
$w$ respectively at $\lambda=0$. The central terms in (3.1.2) contain

$\displaystyle
c_{W_{F,h}}=\frac{2^{2(h-3)}\big{(}(h-1)!\big{)}^{2}}{(2h-3)!!(2h-1)!!}\,,c_{W_{B,h}}=\frac{2^{2(h-3)}(h-2)!h!}{(2h-3)!!(2h-1)!!}\,,c_{Q_{h}}=\frac{2^{2(h-1)}(h-1)!h!}{\big{(}(2h-1)!!\big{)}^{2}}\,.$
(3.6)

Therefore, the OPEs between the first three currents in (3.4) are given by
(3.1.2). The corresponding (anti)commutators can be obtained by using the
procedures described in [19]. See Appendix (B.1).

####  3.1.3 The $q\rightarrow 0$ limit

By adding the bosonic currents and making the contractions with Kronecker
delta (and for fermionic currents we can do similarly) and rescaling them by
$q$ factors,

$\displaystyle
W_{h}\equiv\frac{1}{2}\,(W_{F,h}^{ab}+W_{B,h}^{ab})\,\delta_{ba}$
$\displaystyle\rightarrow$ $\displaystyle q^{h-2}\,W_{h},\qquad
Q_{h+\frac{1}{2}}\equiv\frac{1}{\sqrt{2}}\,Q_{h+\frac{1}{2}}^{ab}\,\delta_{ba}\rightarrow
q^{h+\frac{1}{2}-\frac{3}{2}}\,Q_{h+\frac{1}{2}},$ (3.7)

we obtain the following (anti)commutators

$\displaystyle\commutator\Big{(W_{h_{1}})_{m}}{(W_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W_{h_{1}})_{m}}{(Q_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\anticommutator\Big{(Q_{h_{1}+\frac{1}{2}})_{r}}{(Q_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(-1)^{h_{2}+1}\,(W_{h_{1}+h_{2}})_{r+s}.$
(3.8)

The central terms for $h_{1}=h_{2}=2$ in the first commutator and for
$h_{1}=h_{2}=1$ in the third anticommutator are present but we do not write
them here. Note that by adding the two bosonic currents, the $\frac{1}{q}$
terms are vanishing in Appendix (B.11). In the first commutator, the conformal
weights for $h_{1}$ and $h_{2}$ are even 666 We can denote this bosonic
subalgebra as $w^{K}_{\frac{\infty}{2}}$ algebra.. The anticommutator
satisfies that the conformal weights for $h_{1}$ and $h_{2}$ are both even or
both odd. Then for the former, after we introduce the imaginary number $i$ in
the mode for fermionic current, we can absorb the minus sign on the right hand
side of last relation in (3.8). We will describe the more general
(anti)commutators in (3.15) where there are no restrictions for the weights on
the bosonic and fermionic currents. As done for $SU(K)$ adjoints, we can
multiply the generators of $SO(K)$ properly into the above (anti)commutators
and obtain the corresponding (anti)commutators with $SO(K)$ group indices.

###  3.2  At $\lambda=\frac{1}{4}$

As noticed in [22, 23], there exists a subalgebra generated by

$\displaystyle V^{(h),+}_{\lambda},h=2,4,6,\cdots,\qquad
V^{(h),-}_{\lambda},h=1,3,5,\cdots,$ $\displaystyle
Q^{(h),+}_{\lambda},h=2,4,6,\cdots,\qquad Q^{(h),-}_{\lambda},h=2,4,6,\cdots,$
(3.9)

where the weights for the bosonic currents $V^{(h),\pm}_{\lambda}$ are given
by $h$ and those for the fermionic currents $Q^{(h),\pm}_{\lambda}$ are given
by $(h-\frac{1}{2})$.

For $h_{1},h_{2}$ even and $h_{3}$ odd, there exists an identity on the
structure constants for the particular value $\lambda=\frac{1}{4}$

$\displaystyle\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\bigg{[}\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\pm
S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}\partial_{z}^{h_{1}+h_{2}-h_{3}-k-1}\partial_{w}^{k}$
$\displaystyle\mp\Big{(}S_{F,L}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\pm
S_{B,L}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}\partial_{z}^{k}\partial_{w}^{h_{1}+h_{2}-h_{3}-k-1}\bigg{]}=0\,,$
(3.10)

and for $h_{1},h_{2}$ even and $h_{3}$ even, the similar identity on the
structure constants for the particular value $\lambda=\frac{1}{4}$ is given by

$\displaystyle\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\bigg{[}\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\pm
S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}\partial_{z}^{h_{1}+h_{2}-h_{3}-k-1}\partial_{w}^{k}$
$\displaystyle\pm\Big{(}S_{F,L}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\pm
S_{B,L}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}\partial_{z}^{k}\partial_{w}^{h_{1}+h_{2}-h_{3}-k-1}\bigg{]}=0.$
(3.11)

For example, in the commutator of
$\commutator{\Big{(}V^{(h_{1}),+}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}V^{(h_{2}),+}_{\frac{1}{4}}\Big{)}_{n}}$,
there are $\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+n}$ and
$\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+n}$ on the right hand side, for
any positive integers $h_{1},h_{2}$ and $h_{3}$. For $h_{3}$ odd with two even
$h_{1}$ and $h_{2}$, due to the identity (3.10), the summation over $h_{3}$
with $\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+n}$ for this commutator
runs over only even number. There is no contribution from the $h_{3}$ odd
case. Similarly, for $h_{3}$ even with two even $h_{1}$ and $h_{2}$, due to
the identity (3.11), the summation over $h_{3}$ with
$\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+n}$ for this commutator runs
over only odd number. There is no contribution from the $h_{3}$ even case.
Therefore, we observe that the right hand side of this particular commutator
contains the bosonic fields described by (3.9).

We summarize the whole (anti)commutators in Appendix $C$ described by Appendix
(C.1) and (C.2).

###  3.3  At $\lambda=\frac{1}{2}$

In this case, the conformal weights for $(\beta,\gamma)$ are equal to
$(\frac{1}{2},\frac{1}{2})$ and they for $(b,c)$ are equal to $(1,0)$. Then we
identify $\beta$ with $\gamma$ and denote it by $\psi$ while we denote the
$(b,c)$ by $(\partial\phi,\phi)$ respectively [22, 23]. In this subsection, we
use the previous OPEs (3.1).

By identifying the following relations

$\displaystyle W^{\lambda=\frac{1}{2}}_{F,h}$ $\displaystyle=$ $\displaystyle-
W_{B,h}^{\lambda=0}\,,\qquad
W^{\lambda=\frac{1}{2}}_{B,h}=W^{\lambda=0}_{F,h}\,,\qquad
Q_{h+\frac{1}{2}}^{\lambda=\frac{1}{2}}=(-1)^{h+1}\,Q^{\lambda=0}_{h+\frac{1}{2}}\,,\qquad$
(3.12)

where the currents at $\lambda=0$ are given in (3.2), (3.4), we obtain the
similar (anti)commutator relations by noting the relations (3.12).

###  3.4  At a general $\lambda$

We can also analyze the previous description for the general $\lambda$ by
considering the original $(\beta,\gamma)$, $(b,c)$ system (2.1) 777In this
case we denote the algebra as ${\cal N}=2$ $W_{1+\infty}[\lambda]$ algbra by
following the footnote $2$ of [19], compared to the one in [19]. This notation
holds for the previous subsection with $\lambda=\frac{1}{4}$.. Ler us
introduce the following redefined bosonic and fermionic currents

$\displaystyle W_{h}$ $\displaystyle\equiv$
$\displaystyle\Big{(}W^{\lambda}_{F,\,h}+W^{\lambda}_{B,\,h}\Big{)}\,,\qquad
W^{\hat{A}}_{h}\equiv\Big{(}W^{\lambda,\hat{A}}_{F,\,h}+W^{\lambda,\hat{A}}_{B,\,h}\Big{)}\,,$
$\displaystyle Q_{h+\frac{1}{2}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\,\Big{(}Q^{\lambda}_{h+\frac{1}{2}}+\bar{Q}^{\lambda}_{h+\frac{1}{2}}\Big{)}\,,\qquad
Q_{h+\frac{1}{2}}^{\hat{A}}\equiv\frac{1}{\sqrt{2}}\,\Big{(}Q^{\lambda,\hat{A}}_{h+\frac{1}{2}}+\bar{Q}^{\lambda,\hat{A}}_{h+\frac{1}{2}}\Big{)}\,.$
(3.13)

The reason for the combination of bosonic currents in (3.13) is that when we
consider each OPE between the bosonic and fermionic currents separately, the
lowest order term contains $\frac{1}{q^{2}}$ which diverges as $q\rightarrow
0$ limit. Therefore, in order to eliminate these divergent terms, we should
consider the above addition for the bosonic currents. Furthermore, for the
fermionic currents, we should consider the above combination because in the
anticommutator, the left hand side should contain the same fermionic current.

####  3.4.1 The $q\rightarrow 0$ limit

By taking the following transformations for the currents,

$\displaystyle W_{h}$ $\displaystyle\longrightarrow$ $\displaystyle
q^{h-2}\,W_{h}\,,\qquad W^{\hat{A}}_{h}\longrightarrow
q^{h}\,W^{\hat{A}}_{h}\,,\qquad$ $\displaystyle Q_{h+\frac{1}{2}}$
$\displaystyle\longrightarrow$ $\displaystyle
q^{h-1}Q_{h+\frac{1}{2}}\,,\qquad Q^{\hat{A}}_{h+\frac{1}{2}}\longrightarrow
q^{h+1}Q^{\hat{A}}_{h+\frac{1}{2}},$ (3.14)

we summarize the $q\rightarrow 0$ limit for the nontrivial nine
(anti)commutators from Appendix (D.1) as follows:

$\displaystyle\commutator\Big{(W_{h_{1}})_{m}}{(W_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator{(W_{\,h_{1}})_{m}}{(W_{\,h_{2}}^{\hat{A}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W_{\,h_{1}+h_{2}-2}^{\hat{A}})_{m+n},$
$\displaystyle\commutator{(W_{\,h_{1}}^{\hat{A}})_{m}}{(W_{\,h_{2}}^{\hat{B}})_{n}}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\,i\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{h_{1}+h_{2}-1})_{m+n},$
$\displaystyle\commutator\Big{(W_{h_{1}})_{m}}{(Q_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\hat{A}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}\,(Q_{h_{1}+h_{2}-\frac{3}{2}}^{\hat{A}})_{m+r},$
$\displaystyle\commutator{(W_{h_{1}}^{\hat{A}})_{m}}{(Q_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}\,(Q_{h_{1}+h_{2}-\frac{3}{2}}^{\hat{A}})_{m+r},$
$\displaystyle\commutator{(W_{h_{1}}^{\hat{A}})_{m}}{(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}i\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r},$
$\displaystyle\anticommutator\Big{(Q_{h_{1}+\frac{1}{2}})_{r}}{(Q_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W_{h_{1}+h_{2}})_{r+s},$
$\displaystyle\anticommutator{(Q_{h_{1}+\frac{1}{2}})_{r}}{(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{\hat{A}}_{h_{1}+h_{2}})_{r+s},$ (3.15)

where the weight $h=1,2,3,\cdots$ in $Q_{h+\frac{1}{2}}$ while the weight
$h=1,2,3,\cdots$ in $W_{\,h}$. Note that the weight $h_{1}=1$ in $W_{\,h_{1}}$
cannot appear in the fourth, fifth and sixth relations in (3.15) from Appendix
(D.1). The central terms are ignored. This is other type of $\lambda$ deformed
${\cal N}=1$ supersymmetric $w^{K}_{1+\infty}$ algebra with $U(K)$ symmetry
after $q\rightarrow 0$ limit 888In the construction of [57], there is no
bosonic current from the bosonic fields of weight $1$, we cannot combine it
with the bosonic current from the fermionic fields as in (3.13). This implies
the bosonic subalgebra is given by $w_{\infty}^{K}$ after $q\rightarrow 0$
limit.. Note that the first with $h_{1}=h_{2}=2$, fourth with
$h_{1}=2,h_{2}=1$ and eighth with $h_{1}=h_{2}=1$ relations of (3.15) consist
of the standard ${\cal N}=1$ superconformal algebra with vanishing central
charges [16, 15] 999See also previous works related to the extensions of this
algebra in [61, 62, 63, 64]..

##  4  An extension of Lie superalgebra $PSU(2,2|4)$

The weight-$\frac{1}{2}$ conjugate pairs of symplectic boson fields [65] are
denoted by $(\lambda^{\alpha},\mu_{\alpha}^{\dagger})$ and
$(\mu^{\dot{\alpha}},\lambda_{\dot{\alpha}}^{\dagger})$ where
$\alpha,\dot{\alpha}=1,2$ while four weight-$\frac{1}{2}$ complex fermions are
denoted by $(\psi^{a},\psi_{a}^{\dagger})$ where $a=1,2,3,4$ [25, 26]. The
$\alpha$ and $\dot{\alpha}$ are spinor indices with respect to two different
$SU(2)$’s and $\psi^{a}(\psi_{a}^{\dagger})$ transforms in the
fundamental(antifundamental) representation of $SU(4)$. The conformal
dimension-$\frac{1}{2}$ fields, $(\lambda^{\alpha},\mu_{\alpha}^{\dagger})$
and $(\mu^{\dot{\alpha}},\lambda_{\dot{\alpha}}^{\dagger})$, are bosonic and
they satisfy quasi statistics. See also the work from the worldsheet
description in [66].

Their nontrivial operator product expansions in the left-moving sector of the
worldsheet theory are given by

$\displaystyle\lambda^{\alpha}(z)\,\mu_{\beta}^{\dagger}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)}\,\delta^{\alpha}_{\beta}+\cdots,$
$\displaystyle\mu^{\dot{\alpha}}(z)\lambda_{\dot{\beta}}^{\dagger}(w)$
$\displaystyle=$
$\displaystyle\frac{1}{(z-w)}\,\delta^{\dot{\alpha}}_{\dot{\beta}}+\cdots,$
$\displaystyle\psi^{a}(z)\,\psi_{b}^{\dagger}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)}\,\delta^{a}_{b}+\cdots.$ (4.1)

The components of ambitwistor fields [67] are given by

$\displaystyle
Z^{I}\equiv(\lambda^{\alpha},\mu^{\dot{\alpha}},\psi^{a}),\qquad
Y_{J}\equiv(\mu_{\alpha}^{\dagger},\lambda_{\dot{\alpha}}^{\dagger},\psi_{a}^{\dagger}).$
(4.2)

Then the above OPE (4.1) with (4.2) can be written as

$\displaystyle Z^{I}(z)\,Y_{J}(w)=\frac{1}{(z-w)}\,\delta^{I}_{J}+\cdots.$
(4.3)

By using the quadratic terms

$\displaystyle J^{I}_{\,\,\,J}\equiv Y_{J}\,Z^{I},$ (4.4)

the generators of Lorentz symmetry $\mathcal{L}^{\alpha}_{\,\,\,\beta}$,
$\mathcal{\dot{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta}}$, the generators of $R$
symmetry $\mathcal{R}^{a}_{\,\,\,b}$, the generators of super translations
$\mathcal{Q}^{a}_{\,\,\,\alpha}$, $\mathcal{\dot{Q}}^{\dot{\alpha}}_{\,\,\,a}$
and $\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta}$, the generators of super
conformal boosts $\mathcal{S}^{\alpha}_{\,\,\,a}$,
$\mathcal{\dot{S}}^{a}_{\,\,\,\dot{\alpha}}$ and
$\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta}}$ are described explicitly. There are
also the $U(1)$ hyper charge $\mathcal{B}$, the central charge $\mathcal{C}$
and the dilatation generator $\mathcal{D}$.

We can construct the stress energy tensor which is

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\,\lambda^{\alpha}\partial\mu^{\dagger}_{\alpha}+\mu^{\dot{\alpha}}\partial\lambda^{\dagger}_{\dot{\alpha}}-\psi^{a}\partial\psi^{\dagger}_{a}-\partial\lambda^{\alpha}\mu^{\dagger}_{\alpha}-\partial\mu^{\dot{\alpha}}\lambda^{\dagger}_{\dot{\alpha}}+\partial\psi^{a}\psi^{\dagger}_{a}\,\Big{)}$
(4.5) $\displaystyle\equiv$ $\displaystyle
L^{\alpha}_{\,\,\,\alpha,\,2}+\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\alpha},\,2}+R^{a}_{\,\,\,a,\,2},$

by considering the derivatives. It is obvious that the quadratic terms (4.4)
are weight $1$ primary operators under (4.5). The precise definitions of the
generators are given by the next subsection.

###  4.1 The extension of Lie superalgebra $PSU(2,2|4)$

We can construct the following generalization of the generators in the Lie
superalgebra $PSU(2,2|4)$ together with (4.1), (4.2) and (4.3)

$\displaystyle L^{\alpha}_{\,\,\,\beta,\,h}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\mu^{\dagger}_{\beta}\,\partial^{k}\lambda^{\alpha}\,,\qquad\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h}=\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\lambda^{\dagger}_{\dot{\beta}}\,\partial^{k}\mu^{\dot{\alpha}}\,,$
$\displaystyle R^{a}_{\,\,\,b,\,h}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\psi^{\dagger}_{b}\,\partial^{k}\psi^{a}\,,\qquad
Q^{a}_{\,\,\,\alpha,\,h}=\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\mu^{\dagger}_{\alpha}\,\partial^{k}\psi^{a}\,,$
$\displaystyle\dot{Q}^{\dot{\alpha}}_{\,\,\,a,\,h}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\psi^{\dagger}_{a}\,\partial^{k}\mu^{\dot{\alpha}}\,,\qquad
P^{\dot{\alpha}}_{\,\,\,\beta,\,h}=\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\mu^{\dagger}_{\beta}\,\partial^{k}\mu^{\dot{\alpha}}\,,$
$\displaystyle S^{\alpha}_{\,\,\,a,\,h}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\psi^{\dagger}_{a}\,\partial^{k}\lambda^{\alpha}\,,\qquad\dot{S}^{a}_{\,\,\,\dot{\alpha},\,h}=\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\lambda^{\dagger}_{\dot{\alpha}}\,\partial^{k}\psi^{a}\,,$
$\displaystyle K^{\alpha}_{\,\,\,\dot{\beta},\,h}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{h-1}\,a_{h,k}\,\partial^{h-k-1}\lambda^{\dagger}_{\dot{\beta}}\,\partial^{k}\lambda^{\alpha}\,,$
(4.6)

by allowing the various derivatives to (4.4). The relative coefficients in
(4.6) are determined by the conditions for the quasiprimary operators. They
are given by

$\displaystyle a_{h,k}\equiv(-1)^{k}\binom{h-1}{k}^{2}\,a_{h,0},\qquad
a_{h,0}\equiv\frac{2^{h-3}(h-1)!}{(2h-3)!!}\,q^{h-2}.$ (4.7)

In this normalization of (4.7), $a_{1,0}=\frac{1}{4q}$ 101010 The bosonic
currents can be rewritten as, by rearranging the terms,  $\displaystyle
L^{\alpha}_{\,\,\,\beta,\,h}(z)$ $\displaystyle=$
$\displaystyle(4q)^{h-2}\sum_{i=0}^{h-1}a^{i}(h,\lambda=\tfrac{1}{2})\,\partial_{z}^{h-i-1}\big{(}(\partial^{i}_{z}\mu^{\dagger}_{\beta})\,\lambda^{\alpha}\big{)}(z),$
$\displaystyle\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h}(z)$
$\displaystyle=$
$\displaystyle(4q)^{h-2}\sum_{i=0}^{h-1}a^{i}(h,\lambda=\tfrac{1}{2})\,\partial_{z}^{h-i-1}\big{(}(\partial^{i}_{z}\lambda^{\dagger}_{\dot{\beta}})\,\mu^{\dot{\alpha}}\big{)}(z),$
$\displaystyle R^{a}_{\,\,\,b,\,h}(z)$ $\displaystyle=$
$\displaystyle(4q)^{h-2}\sum_{i=0}^{h-1}a^{i}(h,\lambda+\tfrac{1}{2}=\tfrac{1}{2})\,\partial_{z}^{h-i-1}\big{(}(\partial^{i}_{z}\psi^{\dagger}_{b})\psi^{a}\big{)}(z),$
$\displaystyle P^{\dot{\alpha}}_{\,\,\,\beta,\,h}(z)$ $\displaystyle=$
$\displaystyle(4q)^{h-2}\sum_{i=0}^{h-1}a^{i}(h,\lambda=\tfrac{1}{2})\,\partial_{z}^{h-i-1}\big{(}(\partial^{i}_{z}\mu^{\dagger}_{\beta})\,\mu^{\dot{\alpha}}\big{)}(z)\,,$
$\displaystyle K^{\alpha}_{\,\,\,\dot{\alpha},\,h}(z)$ $\displaystyle=$
$\displaystyle(4q)^{h-2}\sum_{i=0}^{h-1}a^{i}(h,\lambda=\tfrac{1}{2})\,\partial_{z}^{h-i-1}\big{(}(\partial^{i}\lambda^{\dagger}_{\dot{\alpha}})\,\lambda^{\alpha}\big{)}(z)\,.$
Then the current $R^{a}_{\,\,\,b,\,h}$ corresponds to $W_{F,h}^{\lambda=0}$
while others do to $W_{B,h}^{\lambda=\frac{1}{2}}$, up to the group indices.
For the fermionic currents, there exist some degrees of freedom in the overall
normalization factors although the same normalization factors compared to the
bosonic currents are taken in this paper..

Then we obtain the following generators corresponding to the extension of the
Lie superalgebra $PSU(2,2|4)$ from (4.6)

$\displaystyle\mathcal{L}^{\alpha}_{\,\,\,\beta,\,h}$ $\displaystyle=$
$\displaystyle
L^{\alpha}_{\,\,\,\beta,\,h}-\frac{1}{2}\delta^{\alpha}_{\beta}\,L^{\gamma}_{\,\,\,\gamma,\,h}\,,\qquad\dot{\mathcal{L}}^{\dot{\alpha}}_{\,\,\dot{\beta},\,h}=\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\,\dot{L}^{\dot{\gamma}}_{\,\,\,\dot{\gamma},\,h}\,,\qquad\mathcal{R}^{a}_{\,\,\,b,\,h}=R^{a}_{\,\,\,b,\,h}-\frac{1}{4}\delta^{a}_{b}\,R^{c}_{\,\,\,c,\,h}\,,$
$\displaystyle\mathcal{Q}^{a}_{\,\,\,\alpha,\,h}$ $\displaystyle=$
$\displaystyle
Q^{a}_{\,\,\,\alpha,\,h}\,,\qquad\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h}=\dot{Q}^{\dot{\alpha}}_{\,\,\,a,\,h}\,,\qquad\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta,\,h}=P^{\dot{\alpha}}_{\,\,\,\beta,\,h}\,,$
$\displaystyle\mathcal{S}^{\alpha}_{\,\,\,a,\,h}$ $\displaystyle=$
$\displaystyle
S^{\alpha}_{\,\,\,a,\,h}\,,\qquad\dot{\mathcal{S}}^{a}_{\,\,\,\dot{\alpha},\,h}=\dot{S}^{a}_{\,\,\,\dot{\alpha},\,h}\,,\qquad\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h}=K^{\alpha}_{\,\,\,\dot{\beta},\,h},$
$\displaystyle\mathcal{B}_{h}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}L^{\alpha}_{\,\,\,\alpha,\,h}+\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\alpha},\,h}\big{)}\,,\mathcal{C}_{h}=\frac{1}{2}\big{(}L^{\alpha}_{\,\,\,\alpha,\,h}+\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\alpha},\,h}+R^{a}_{\,\,\,a,\,h}\big{)}\,,\mathcal{D}_{h}=\frac{1}{2}\big{(}L^{\alpha}_{\,\,\,\alpha,\,h}-\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\alpha},\,h}\big{)}\,.$
(4.8)

Note that we have $\mathcal{R}^{a}_{\,\,\,a,\,h}=0$ from the third relation of
(4.8). Furthermore, there are following currents by contractions in each
$SU(2)$, $SU(2)$ and $SU(4)$ index respectively

$\displaystyle\mathcal{U}_{h}$ $\displaystyle=$ $\displaystyle
L^{\gamma}_{\,\,\,\gamma,\,h}\,,\qquad\dot{\mathcal{U}}_{h}=\dot{L}^{\dot{\gamma}}_{\,\,\,\dot{\gamma},\,h}\,,\qquad\mathcal{V}_{h}=R^{c}_{\,\,\,c,\,h}\,,$
(4.9)

which can be written in terms of the last three generators in (4.8).

###  4.2  The construction of bosonic current

Because the stress energy tensor is given by (4.5), we consider the following
extension of stress energy tensor, by adding the currents in (4.9),

$\displaystyle W_{h}\equiv
L^{\alpha}_{\,\,\,\alpha,\,h}+\dot{L}^{\dot{\alpha}}_{\,\,\,\dot{\alpha},\,h}+R^{a}_{\,\,\,a,\,h}\,=2\,\mathcal{C}_{h}.$
(4.10)

Then its commutator relation satisfies 111111 According to Appendix $B$, there
exists a relation  $\displaystyle p_{B}^{h_{1},h_{2},h}(m,n,\tfrac{1}{2})$
$\displaystyle=p_{F}^{h_{1},h_{2},h}(m,n,0)\,=\frac{1}{2(h+1)!}\,\phi_{h+2}^{h_{1},h_{2}}(0,1)\,N^{h_{1},h_{2}}_{h}(m,n).$
In particular, the mode dependent function leads to  $\displaystyle
N^{h_{1},h_{2}}_{-1}(m,n)$ $\displaystyle=1\,,\qquad N^{h_{1},h_{2}}_{0}(m,n)$
$\displaystyle=2\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,.$ Moreover,
$\displaystyle\phi_{-1+2}^{h_{1},h_{2}}(0,1)$
$\displaystyle=1\,,\qquad\phi_{0+2}^{h_{1},h_{2}}(0,1)$ $\displaystyle=1\,.$
This implies  $\displaystyle
p_{B}^{h_{1},h_{2},-1}(m,n,\tfrac{1}{2})=\frac{1}{2},\qquad
p_{B}^{h_{1},h_{2},0}(m,n,\tfrac{1}{2})=\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}.$

$\displaystyle\commutator{(W_{h_{1}})_{m}}{(W_{h_{2}})_{n}}$
$\displaystyle=\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(W_{h_{1}+h_{2}-2h-2})_{m+n}\,.$
(4.11)

Let us write down some of the OPEs as follows:

$\displaystyle W_{1}(z)\,W_{2}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{2}}\,W_{1}(w)+\cdots\,,$ $\displaystyle
W_{1}(z)\,W_{3}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{2}}\,2\,W_{2}(w)+\cdots\,,$ $\displaystyle
W_{2}(z)\,W_{2}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{2}}\,2\,W_{2}(w)+\frac{1}{(z-w)}\,\partial
W_{2}(w)+\cdots\,,$ $\displaystyle W_{2}(z)\,W_{3}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{4}}\,16q^{2}\,W_{1}(w)+\frac{1}{(z-w)^{2}}\,3\,W_{3}(w)+\frac{1}{(z-w)}\,\partial
W_{3}(w)+\cdots\,,$ $\displaystyle W_{3}(z)\,W_{3}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{4}}\,64q^{2}\,W_{2}(w)+\frac{1}{(z-w)^{3}}\,32q^{2}\,\partial
W_{2}(w)$ (4.12) $\displaystyle+$
$\displaystyle\frac{1}{(z-w)^{2}}\,\bigg{[}\,4\,W_{4}+\frac{48\,q^{2}}{5}\,\partial^{2}W_{2}\,\bigg{]}(w)$
$\displaystyle+$ $\displaystyle\frac{1}{(z-w)}\,\bigg{[}\,2\,\partial
W_{4}+\frac{32\,q^{2}}{15}\,\partial^{3}W_{2}\,\bigg{]}(w)+\cdots.$

We can check these OPEs (4.12) are the same as the equation $(3.5)$ of [19]
with $\lambda=0$ and vanishing central terms: $W_{1+\infty}[\lambda=0]$
algebra. In general, when we take $\lambda\rightarrow 0$ in the first equation
of Appendix $(D.1)$ of [19], the corresponding commutator without a central
term becomes the one in (4.11).

###  4.3  The OPEs between the bosonic current and other generators

It is straightforward to calculate the following commutator relations with
(4.10)

$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{L}^{\alpha}_{\,\,\,\beta,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{L}^{\alpha}_{\,\,\,\beta,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\dot{\mathcal{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\dot{\mathcal{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{R}^{a}_{\,\,\,b,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{R}^{a}_{\,\,\,b,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{\,h_{1}})_{m}}{(\mathcal{S}^{\alpha}_{\,\,\,a,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{S}^{\alpha}_{\,\,\,a,\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{\,h_{1}})_{m}}{(\dot{\mathcal{S}}^{a}_{\,\,\,\dot{\alpha},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\dot{\mathcal{S}}^{a}_{\,\,\,\dot{\alpha},\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{\,h_{1}})_{m}}{(\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{B}_{\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle-q^{2(h_{1}-2)}\,2\,\delta_{h_{1},h_{2}}\,c_{W_{F,h_{1}}}(m)\,\delta_{m+n}$
$\displaystyle+$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{B}_{h_{1}+h_{2}-2h-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{D}_{\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h=0}^{(h_{1}+h_{2}-3)/2}q^{2h}\,p_{B}^{h_{1},h_{2},2h}(m,n,\tfrac{1}{2})\,(\mathcal{D}_{h_{1}+h_{2}-2h-2})_{m+n}\,,$
(4.13)

where the central term appearing in (4.13) together with (3.6) is given by

$\displaystyle c_{W_{F,h}}(m)$
$\displaystyle=\frac{2^{2(h-3)}((h-1)!)^{2}}{(2h-3)!!(2h-1)!!}[m+h-1]_{2h-1}\,.$
(4.14)

Note that the structure constants (mode dependent terms) in (4.13) are given
by the $N$ function according to the footnote 11. We can obtain the
corresponding OPEs of (4.13) by using the procedure in [19].

###  4.4  The $q\rightarrow 0$ limit

We obtain the $q\rightarrow 0$ limit for the commutators in (4.13) by
rescaling of $q^{h-2}$ for the bosonic currents and $q^{h-1}$ for the
fermionic currents with the help of the footnote 11

$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{L}^{\alpha}_{\,\,\,\beta,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{L}^{\alpha}_{\,\,\,\beta,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{\dot{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{\dot{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{R}^{a}_{\,\,\,b,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{R}^{a}_{\,\,\,b,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{S}^{\alpha}_{\,\,\,a,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{S}^{\alpha}_{\,\,\,a,\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{\dot{S}}^{a}_{\,\,\,\dot{\alpha},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{\dot{S}}^{a}_{\,\,\,\dot{\alpha},\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{B}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{B}_{h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(W_{h_{2}})_{n}}$ $\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W_{h_{1}+h_{2}-2})_{m+n}\,,$
$\displaystyle\commutator{(W_{h_{1}})_{m}}{(\mathcal{D}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(\mathcal{D}_{h_{1}+h_{2}-2})_{m+n}\,,$
(4.15)

where the central term appearing in the third relation from the below in
(4.15) is ignored 121212The bosonic subalgebra $w_{1+\infty}$ algebra appears
in the second relation from the below in (4.15)..

Furthermore, the anticommutators are given by

$\displaystyle\anticommutator{(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{1}})_{m}}{(\dot{\mathcal{Q}}^{\dot{\beta}}_{\,\,\,b,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\,\delta^{a}_{b}\,(\mathcal{P}^{\dot{\beta}}_{\,\,\,\alpha,\,h_{1}+h_{2}-1})_{m+n},$
$\displaystyle\anticommutator{(\mathcal{Q}^{a}_{\,\,\,\alpha,\,h_{1}})_{m}}{(\mathcal{S}^{\beta}_{\,\,\,b,\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\Bigg{(}\delta^{\beta}_{\alpha}\,(\mathcal{R}^{a}_{\,\,\,b,\,h_{1}+h_{2}-1})_{m+n}+\delta^{a}_{b}\,(\mathcal{L}^{\beta}_{\,\,\,\alpha,\,h_{1}+h_{2}-1})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{a}_{b}\delta^{\beta}_{\alpha}\,\bigg{[}\frac{1}{2}\,(\mathcal{U}_{h_{1}+h_{2}-1})_{m+n}+\frac{1}{4}\,(\mathcal{V}_{h_{1}+h_{2}-1})_{m+n}\bigg{]}\Bigg{)},$
$\displaystyle\anticommutator{(\dot{\mathcal{Q}}^{\dot{\alpha}}_{\,\,\,a,\,h_{1}})_{m}}{(\dot{\mathcal{S}}^{b}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\Bigg{(}\,\delta^{\dot{\alpha}}_{\dot{\beta}}\,(\mathcal{R}^{b}_{\,\,\,a,\,h_{1}+h_{2}-1})_{m+n}+\delta^{b}_{a}\,(\dot{\mathcal{L}}^{\dot{\alpha}}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-1})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{b}_{a}\delta^{\dot{\alpha}}_{\dot{\beta}}\,\bigg{[}\frac{1}{2}\,(\dot{\mathcal{U}}_{h_{1}+h_{2}-1})_{m+n}+\frac{1}{4}\,(\mathcal{V}_{h_{1}+h_{2}-1})_{m+n}\bigg{]}\Bigg{)},$
$\displaystyle\anticommutator{(\mathcal{S}^{\alpha}_{\,\,\,a,\,h_{1}})_{m}}{(\dot{\mathcal{S}}^{b}_{\,\,\,\dot{\beta},\,h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\,\delta^{b}_{a}\,(\mathcal{K}^{\alpha}_{\,\,\,\dot{\beta},\,h_{1}+h_{2}-1})_{m+n},$
(4.16)

where the central terms appearing in (4.16), which can be seen from Appendix
$E$ and contain the term (4.14), are ignored.

Other commutators after $q\rightarrow 0$ limit with appropriate rescalings for
the modes of the currents can be determined from the final results in Appendix
$E$.

##  5  The application in the celestial conformal field theory

According to the celestial holography, we present the soft current algebra by
combining the algebras obtained in previous sections with the insertion of the
helicity and propose the corresponding bulk theory based on the ${\cal N}=1$
supergravity theory.

###  5.1  The soft current algebra in section $2$

The algebra presented in (2.22) appears also in [12] where there is no
$\lambda$ deformation in the two dimensional conformal field theory. For a
given ${\cal N}=2$ supersymmetric $W^{K}_{1+\infty}[\lambda]$ algebra [19],
via topological twisting [21], the corresponding ${\cal N}=1$ supersymmetric
topological $W^{K}_{\infty}[\lambda]$ 131313 In an abstract, we use a
simplified notation for this as $W_{\infty}$. is obtained in Appendix $A$. By
using the rescalings for the modes (2.21), the previous algebra (2.22) is
determined explicitly. As discussed in [12, 13], according to the celestial
holography, the corresponding OPEs (or commutators) in the ${\cal N}=1$
supersymmetric Einstein Yang-Mills theory are obtained in [11], even for the
presence of the deformation parameter $\lambda$. The soft gravitons correspond
to the bosonic $SU(K)$ singlet current while the soft gluons do the bosonic
$SU(K)$ adjoint current. Similarly, the soft gravitinos correspond to the
fermionic $SU(K)$ singlet current while the soft gluinos do the fermionic
$SU(K)$ adjoint current.

By writing down the helicities explicitly, the soft current algebra is
summarized by

$\displaystyle\commutator{(\tilde{W}^{\lambda,+2}_{h_{1}})_{m}}{(\tilde{W}^{\lambda,+2}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(\tilde{W}^{\lambda,+2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,+2}_{h_{1}})_{m}}{(\tilde{W}^{\lambda,\hat{A},+1}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(\tilde{W}^{\lambda,\hat{A},+1}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A},+1}_{h_{1}})_{m}}{(\tilde{W}^{\lambda,\hat{B},+1}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle-\frac{i}{4}f^{\hat{A}\hat{B}\hat{C}}(\tilde{W}^{\lambda,\hat{C},+1}_{h_{1}+h_{2}-1})_{m+n},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,+2}_{h_{1}})_{m}}{(Q^{\lambda,+\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda,+\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,+2}_{h_{1}})_{m}}{(Q^{\lambda,\hat{A},+\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda,\hat{A},+\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A},+1}_{h_{1}})_{m}}{(Q^{\lambda,+\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}h_{2}\,m-(h_{1}-1)r+\frac{h_{1}-1}{2}\Big{)}(Q^{\lambda,\hat{A},+\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator{(\tilde{W}^{\lambda,\hat{A},+1}_{h_{1}})_{m}}{(Q^{\lambda,\hat{B},+\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle-\frac{i}{4}f^{\hat{A}\hat{B}\hat{C}}(Q^{\lambda,\hat{C},+\frac{1}{2}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}.$
(5.1)

The analysis for the first three commutators is done in [6]. See also the soft
symmetry analysis from the Carrollian amplitudes (which comes from other flat
space holography) studied in [68].

###  5.2 The soft current algebra in subsection $3.1$

For a given ${\cal N}=2$ supersymmetric $W^{K}_{1+\infty}[\lambda]$ algebra
[19], by introducing the real free fields at fixed $\lambda=0$, the
corresponding ${\cal N}=1$ supersymmetric $W_{1+\infty}[\lambda=0]$ (or
$W_{\frac{\infty}{2}}$) is obtained in Appendix $B$. By using the rescalings
for the modes (3.7), the previous algebra (3.8) is obtained explicitly.
Compared to the one in (2.22), there exists an anticommutator between the
modes for the fermionic current. The analysis for the soft current algebra we
will not present them in this paper can be done similarly by following the
procedure in next subsection.

###  5.3  Other soft current algebra in subsection $3.4$

The ${\cal N}=2$ supersymmetric $W^{K}_{1+\infty}[\lambda]$ algebra [19]
provides the corresponding ${\cal N}=1$ supersymmetric
$W^{K}_{1+\infty}[\lambda]$ by using the relations (3.13) 141414 We use a
simplified notation for this as $W_{1+\infty}[\lambda]$ in an abstract. . By
using the rescalings for the modes (3.14), the algebra (3.15) is determined
explicitly. In this case, there are no restrictions on the weights for the
modes of bosonic currents compared to the one (3.8) where the weights for
those are even. Similar behavior for the fermionic currents occurs. We expect
that in the celestial conformal field theory, the anticommutator should be
described by the soft gravitinos with the helicity $\pm\frac{3}{2}$, and the
soft photinos (or the soft spin $\frac{1}{2}$ field) with the helicity
$\pm\frac{1}{2}$ 151515The analysis on this subsection is based on the
intensive discussion for the various aspects on the celestial holography with
M. Pate during CA’s visit at NYU in Sept., 2022-Feb., 2023..

####  5.3.1  The ${\cal N}=1$ supergravity theory

Let us consider the (simple) ${\cal N}=1$ supergravity found in [69]. See also
[70]. The corresponding soft algebra, by putting the helicities in the
superscript (and recalling the construction of [71, 72] in the context of the
symplecton algebra), can be described as

$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(W^{\pm 2}_{h_{2}})_{n}}$
$\displaystyle=$ $\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{\pm
2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(Q^{\pm\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{\pm\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{3}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-2}_{h_{1}+h_{2}})_{r+s}.$ (5.2)

From the one of the gravitational action(or Lagrangian)
$\frac{1}{\kappa^{2}}\,e\,R$ where $e\equiv\mbox{det}\,e^{a}_{\mu}$ for
vierbein $e^{a}_{\mu}$ and $R$ is a scalar curvature [69], the linear
gravitational coupling $\kappa$ term contains three gravitons with two
derivatives [73]. Because there exist two derivatives, the dimension of the
three point vertex is given by $d_{V}=3+2=5$ where $3$ comes from each
dimension of graviton [10]. Then the sum of helicities becomes $(d_{V}-3)$
[74]. This implies that the sum of helicities should be equal to $2$.
Therefore, the first commutator in (5.2) leads to the helicities $(+2,\pm
2,\mp 2)$ for three gravitons when $d_{V}=5$. Let us emphasize that the
helicity on the right hand side in (anti)commutators appears negatively. All
the possible helicities from $(\pm 2,\pm\frac{3}{2})$ satisfying $d_{V}=5$ are
listed in (5.2) 161616 We can check this by using the following mathematica
calculation:  $\displaystyle{\tt Clear[SoT]}$ $\displaystyle{\tt
SoT[x_{,}y_{,}z_{]}:=True/;x+y+z==2}$ $\displaystyle{\tt
SoT[x_{,}y_{,}z_{]}:=False/;x+y+z!=2}$ $\displaystyle{\tt
Do[If[SoT[x,y,z],Print[{x,y,z}]],\\{x,\\{-2,2,-\frac{3}{2},\frac{3}{2}\\}\\},\\{y,\\{-2,2,-\frac{3}{2},\frac{3}{2}\\}\\},\\{z,\\{-2,2,-\frac{3}{2},\frac{3}{2}\\}\\}]}$
Here, $x$ $y$ stand for the helicities appearing on the left hand side while
$z$ stands for the helicity appearing on the right hand side of
(anti)commutator. Among nine outputs, the independent five outputs occur as in
(5.2)..

From the next gravitational action
$\epsilon^{\mu\nu\rho\sigma}\,\bar{\psi}_{\mu}\,\gamma_{5}\,\gamma_{\nu}\,D_{\rho}\,\psi_{\sigma}$
where $\psi_{\mu}$ is a spin $\frac{3}{2}$ Majorana spinor and $D_{\mu}$ is
the gravitationally covariant derivative [69], it is known that $\gamma_{5}$
is a constant while $\gamma_{\nu}$ contains the above vierbein [75] where the
vierbein $e^{a}_{\mu}$ is expanded around the flat Minkowski spacetimes [76].
The linear $\kappa$ term contains a graviton. Because there exists a single
derivative, the dimension of the three point vertex is given by
$d_{V}=\frac{3}{2}+\frac{3}{2}+1+1=5$ where $\frac{3}{2}$ comes from each
dimension of gravitino and $1$ does from the dimension of graviton. Then we
can write down the second commutator relation in (5.2) and the helicities are
given by $(+2,\pm\frac{3}{2},\mp\frac{3}{2})$ for graviton and two gravitinos.
Moreover, the helicities in the last relation in (5.2) are given by
$(+\frac{3}{2},-\frac{3}{2},+2)$ for two gravitinos and graviton. See also
[11] for the presence of this kind of interaction 171717 Recently, in [77],
the three point interactions corresponding to the first two commutators of
(5.2) are calculated. The first one comes from the product of two Yang-Mills
vertices involving the gluons while the second one comes from the product of a
super Yang-Mills vertex involving two gluinos and one gluon and a Yang-Mills
vertex involving gluons together with various polarization tensors [78]..

####  5.3.2  The ${\cal N}=1$ supersymmetric Maxwell Einstein theory

It is known that the spins $(2,\frac{3}{2})$ gauge multiplet for supergravity
[69] (See also [70]) is coupled to the spins $(1,\frac{1}{2})$ vector
multiplet in the Maxwell Einstein theory in [79]. This can be also obtained
from the simplest ${\cal N}=3$ supergravity theory [80] by consistent
truncation.

We summarize the soft current algebra with the explicit helicities appearing
in the superscript of the modes for the currents

$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(W^{\pm 1}_{h_{2}})_{n}}$
$\displaystyle=$ $\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{\pm
1}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(W^{\pm 2}_{h_{2}})_{n}}$
$\displaystyle=$ $\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{\pm
2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+1}_{h_{1}})_{m}}{(W^{-1}_{h_{2}})_{n}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{-2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(Q^{\pm\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{\pm\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(Q^{\pm\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{\pm\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{+1}_{h_{1}})_{m}}{(Q^{+\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{+\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{+1}_{h_{1}})_{m}}{(Q^{-\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{-\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{3}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-2}_{h_{1}+h_{2}})_{r+s},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{1}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-2}_{h_{1}+h_{2}})_{r+s},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{3}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-1}_{h_{1}+h_{2}})_{r+s}.$ (5.3)

The second, fourth, and eighth relations of (5.3) can be analyzed similarly as
done before.

From the one of the matter action
$e\,g^{\mu\rho}\,g^{\nu\sigma}\,F_{\mu\nu}\,F_{\rho\sigma}$ [79] with
$F_{\mu\nu}\equiv\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}$ and
$e\equiv\mbox{det}\,e^{a}_{\mu}$, we observe the interaction between the
graviton and two photons by expanding the vierbein $e^{a}_{\mu}$ around the
flat Minkowski spacetimes [76] as before and noticing that the linear term in
the gravitational coupling $\kappa$ contains the graviton. Because there exist
two derivatives, the dimension of the three point vertex is given by
$d_{V}=3+2=5$ where $3$ comes from each dimension of graviton and photon [10].
Then sum of helicities becomes $(d_{V}-3)$ [74]. This implies that the sum of
helicities should be equal to $2$. Therefore, we conclude that the first
commutator in (5.3) shows the helicities $(+2,\pm 1,\mp 1)$ for graviton,
photon and photon. Note that the helicity on the right hand side in
(anti)commutators appears negatively. The third commutator in (5.3) can be
analyzed similarly and the helicities are given by $(+1,-1,+2)$ for two
photons and graviton. See also [81] for the presence of this kind of
interaction.

From the next matter action $e\,\bar{\lambda}\,\gamma^{\mu}\,D_{\mu}\,\lambda$
where $\lambda$ is a Majorana spin $\frac{1}{2}$ field [79], there exists a
single derivative, and $d_{V}=\frac{3}{2}+\frac{3}{2}+1+1=5$ where
$\frac{3}{2}$ comes from each dimension of photino 181818In [75], the
terminology ’neutrino’ is used. and $1$ does from the dimension of graviton as
before. The fifth commutator relation in (5.3) contains the helicities by
$(+2,\pm\frac{1}{2},\mp\frac{1}{2})$ for graviton and two photinos. Moreover,
the helicities in the ninth relation in (5.3) are given by
$(+\frac{1}{2},-\frac{1}{2},+2)$ for two photinos and graviton. See also [11]
for the presence of this kind of interaction.

From the last matter action
$e\,\kappa\,\bar{\psi}_{\mu}\,\gamma^{\alpha}\,\gamma^{\beta}\,\gamma^{\mu}\,\lambda\,F_{\alpha\beta}$
at the level of linear $\kappa$ [79], there exists a single derivative, and
$d_{V}=\frac{3}{2}+\frac{3}{2}+1+1=5$ where $\frac{3}{2}$ comes from each
dimension of gravitino, and photino and $1$ does from the dimension of photon.
In this case, the vierbein plays the role of Kronecker delta appearing in the
lowest order of $\kappa$($\kappa$ independent term) because this action has a
linear $\kappa$ term. The sixth commutator relation in (5.3) contains the
helicities by $(+1,+\frac{3}{2},-\frac{1}{2})$ for photon, gravitino and
photino. Moreover, the helicities in the seventh relation in (5.3) are given
by $(+1,-\frac{1}{2},+\frac{3}{2})$ for photon, photino and gravitino.
Finally, the last anticommutator relation in (5.3) contains the helicities by
$(+\frac{3}{2},-\frac{1}{2},+1)$ for gravitino, photino and the photon. See
also [81, 11] for the presence of this kind of interaction. Note that some
possible helicities from $(\pm 2,\pm\frac{3}{2},\pm 1,\pm\frac{1}{2})$
satisfying $d_{V}=5$ are listed in (5.3) and the other possible helicities
$(+\frac{1}{2},+\frac{1}{2},+1)$ and $(+\frac{3}{2},+\frac{3}{2},-1)$ are not
present 191919This can be checked by using the similar calculation done in the
footnote 16 by further considering the helicities $\pm 1$ and
$\pm\frac{1}{2}$. in this ${\cal N}=1$ supersymmetric Maxwell Einstein theory
202020In [82], the Yang-Mills theory is coupled to the simple ${\cal N}=1$
supergravity. It is straightforward to describe the corresponding soft algebra
from the second, third, fifth, sixth, seventh and ninth equations of (3.15) as
done in the ${\cal N}=1$ Maxwell Einstein theory..

####  5.3.3  The ${\cal N}=1$ supergravity theory coupled to the scalar
multiplet

The globally supersymmetric multiplet contains a scalar field $A$ (denoted by
the helicity $+0$), a pseudoscalar field $B$ (denoted by the helicity $-0$)
and a real spin $\frac{1}{2}$ field $\chi$. The full locally supersymmetric
Lagrangian is described in [83]. We describe the corresponding soft algebra as
follows:

$\displaystyle\commutator\Big{(W^{\pm 0}_{h_{1}})_{m}}{(W^{\pm
0}_{h_{2}})_{n}}$ $\displaystyle=$
$\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{-2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{\pm 0}_{h_{1}})_{m}}{(W^{+2}_{h_{2}})_{n}}$
$\displaystyle=$ $\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{\pm
0}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(W^{\pm 2}_{h_{2}})_{n}}$
$\displaystyle=$ $\displaystyle\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}(W^{\pm
2}_{h_{1}+h_{2}-2})_{m+n},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(Q^{\pm\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{\pm\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{+2}_{h_{1}})_{m}}{(Q^{\pm\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$
$\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{\pm\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{\pm
0}_{h_{1}})_{m}}{(Q^{+\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{r}}$ $\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{-\frac{3}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\commutator\Big{(W^{\pm
0}_{h_{1}})_{m}}{(Q^{+\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{r}}$ $\displaystyle=$
$\displaystyle\Big{(}(h_{2}-\frac{1}{2})m-(h_{1}-1)r\Big{)}(Q^{-\frac{1}{2}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{3}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-2}_{h_{1}+h_{2}})_{r+s},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{1}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{-\frac{1}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{-2}_{h_{1}+h_{2}})_{r+s},$
$\displaystyle\anticommutator\Big{(Q^{+\frac{1}{2}}_{h_{1}+\frac{1}{2}})_{r}}{(Q^{+\frac{3}{2}}_{h_{2}+\frac{1}{2}})_{s}}$
$\displaystyle=$ $\displaystyle 2\,(W^{\pm 0}_{h_{1}+h_{2}})_{r+s}.$ (5.4)

The third, fourth, fifth, eighth and ninth relations of (5.4) can be analyzed
similarly as done before. Of course, the role of spin $\frac{1}{2}$ fields
appearing in (5.3) and (5.4) is different from each other.

From $e\,g^{\mu\nu}\,\partial_{\mu}\,A\,\partial_{\nu}\,A$ and
$e\,g^{\mu\nu}\,\partial_{\mu}\,B\,\partial_{\nu}\,B$ [83], we observe the
interaction between the graviton and two scalars by expanding the vierbein
$e^{a}_{\mu}$ around the flat Minkowski spacetimes and noticing that the
linear term in the gravitational coupling $\kappa$ contains the graviton as
before. Because there exist two derivatives, the dimension of the three point
vertex is given by $d_{V}=5$. Then sum of helicities becomes $(d_{V}-3)$. This
implies that the sum of helicities should be equal to $2$. Therefore, we
conclude that the first commutator in (5.4) shows the helicities $(\pm 0,\pm
0,+2)$ for scalar, scalar and graviton. The second commutator can be analyzed
similarly.

From $e\,\kappa\,\bar{\psi}_{\mu}\,\not{\partial}\,A\,\gamma^{\mu}\,\chi$ and
$e\,\kappa\,\bar{\psi}_{\mu}\,\gamma^{5}\,\not{\partial}\,B\,\gamma^{\mu}\,\chi$,
there exists a single derivative, and $d_{V}=\frac{3}{2}+\frac{3}{2}+1+1=5$
where $\frac{3}{2}$ comes from each dimension of gravitino, and spin
$\frac{1}{2}$ field $\chi$ and $1$ does from the dimension of scalar. In this
case, the vierbein plays the role of Kronecker delta. The sixth commutator
relation in (5.4) contains the helicities by $(\pm
0,+\frac{1}{2},+\frac{3}{2})$ for scalar, spin $\frac{1}{2}$ field and
gravitino. Moreover, the helicities in the seventh relation in (5.4) are given
by $(\pm 0,+\frac{3}{2},+\frac{1}{2})$ for scalar, gravitino and spin
$\frac{1}{2}$ field. The last anticommutator can be described similarly. Note
that all the possible helicities from $(\pm
2,\pm\frac{3}{2},\pm\frac{1}{2},0)$ satisfying $d_{V}=5$ are listed in (5.4)
with the help of the footnote 16 212121 We can check that the soft current
algebras in (5.2), (5.3) and (5.4) can arise in the soft operators [28, 29]
after ignoring the $R$ symmetry $SU(8)_{R}$ indices, along the line of [84].
That is, we do not have their equation $(2.23)$ of [29] which does not have
$\frac{1}{(z-w)}$ term, the first equation of Appendix $(A.1)$ having the
helicity $(+\frac{3}{2},+\frac{3}{2},-1)$, the first equation of Appendix
$(A.2)$ with the helicity $(+1,+1,0)$, the first three equations of Appendix
$(A.3)$ where the helicity is given by $(+\frac{1}{2},+\frac{1}{2},+1)$, the
second equation of Appendix $(A.9)$ having the helicity
$(+\frac{3}{2},-1,+\frac{3}{2})$, the first equation of Appendix $(A.12)$
where the helicity is $(+1,+\frac{1}{2},+\frac{1}{2})$ and the first equations
of Appendix $(A.13)$ with the helicity $(+1,0,+1)$ in our case, but we do have
their remaining equations. We can find the similar interaction with the
helicity $(+1,+1,0)$ in [85, 86]. All the possible helicities from $(\pm
2,\pm\frac{3}{2},\pm 1,\pm\frac{1}{2},0)$ satisfying $d_{V}=5$ can be obtained
by using the procedure presented in the footnote 16. The above helicities
$(+\frac{3}{2},+\frac{3}{2},-1)$ corresponding to the interaction which is a
generalization of $e\,\kappa\,\bar{\psi}_{\mu}\,F^{\mu\,\nu}\,\psi_{\nu}$,
$(+\frac{1}{2},+\frac{1}{2},+1)$ corresponding to the interaction which is a
generalization of
$e\,\kappa\,\bar{\chi}\,\sigma^{\mu\,\nu}\,\chi\,F_{\mu\,\nu}$ and $(+1,+1,0)$
corresponding to the interaction which is a generalization of
$e\,\kappa\,A\,g^{\mu\,\rho}\,g^{\nu\,\sigma}\,F_{\mu\,\nu}\,F_{\rho\,\sigma}$
satisfy the $d_{V}=5$ condition but do not appear in the simple supergravity
theory coupled to the matter of this paper. We expect that these interactions
will appear in more supersymmetric (or extended) supergravity theories by
putting the appropriate $R$ symmetry indices on the fields. The various OPEs
between the operators having the Euler Beta function can be determined
explicitly..

###  5.4  Towards the soft current algebra in section $4$

The Lie superalgebra $PSU(2,2|4)$ appears in [87, 88] when we construct the
conformal supergravity by gauging the superconformal symmetry of flat space
theories [89]. See also [90]. The physical fields are the graviton
corresponding to the generator $\mathcal{P}^{\dot{\alpha}}_{\,\,\,\beta}$,
four gravitinos corresponding to the generators
$\mathcal{Q}^{a}_{\,\,\,\alpha}$ and $SU(4)$ vectors corresponding to the $R$
symmetry generators $\mathcal{R}^{a}_{\,\,\,b}$. Note that in the lower
supersymmetric case with Lie superalgebra $SU(2,2|{\cal N}=1,2,3)$, there
exists an additional $U(1)$ vector corresponding to the generator
$\mathcal{R}^{a}_{\,\,\,a}$. See also [91, 92, 93].

From the second relation from the last in (4.15), we observe the commutator
relation between the soft gravitons with helicities $(+2,\pm 2)$ producing the
soft gravitons with helicity $\mp 2$. From the remaining relations in (4.15),
the corresponding soft algebra can be read off directly. For the
anticommutators, we can take a summation over the spinor indices in the second
relation of (4.16) by multiplying $\delta^{\alpha}_{\beta}$. Then on the right
hand side, there exist the mode for $\mathcal{R}^{a}_{\,\,\,b,h_{1}+h_{2}-1}$,
the mode for $W_{h_{1}+h_{2}-1}$ and the mode for the dilatation generator
${\cal D}_{h_{1}+h_{2}-1}$.

Moreover, when we consider the commutator between
$(\mathcal{Q}^{a}_{\,\,\,\alpha,h_{1}})_{m}$ and $({\cal D}_{h_{2}})_{n}$, the
$\frac{1}{q^{2}}$ term arises on the right hand side of this commutator. At
the moment, we cannot remove this divergent term. One way to obtain the closed
algebra under the particular limit for the parameter is to consider $\lambda$
deformed version around (4.1) by introducing $\lambda$ dependent weights of
symplectic bosons and complex fermions. It is an open problem to observe
whether the presence of $\lambda$ affects the above anticommutator and leads
to the well defined closed algebra. There is other possibility for the overall
normalizations for the fermionic currents to remove the divergent terms.

##  6  Conclusions and outlook

The new supersymmetric algebras obtained in this paper are given by the ones
in Appendices $A,B,C$ and $E$. Note that there exists a nonsupersymmetric
algebra in Appendix $C$. The corresponding $q\rightarrow 0$ limits for
Appendices $A,B,D$ and $E$ are considered. The algebra corresponding to
Appendix $D$ appears in [19]. The dual descriptions for these algebras in
Appendices $A$ and $D$ (or $B$ or $C$) under the $q\rightarrow 0$ limit are
obtained in section $5$: See the equations (5.1), (5.2), (5.3) and (5.4).

In the subsection $5.3$, some ${\cal N}=1$ supergravity theories are
interpreted as the duals for the soft current algebras found in this paper in
the celestial conformal field theory. We expect that as we increase the number
of supersymmetry, it is more possible to cover the interactions we do not
consider in this paper. See the footnote 21. It is known that there exist
${\cal N}=2$ supergravity in [94], ${\cal N}=3$ supergravity in [95, 80], and
${\cal N}=4$ supergravity in [96, 97, 98, 99]. It is natural to study the
${\cal N}=4$ supersymmetric $W_{1+\infty}[\lambda]$ algebra [100, 101, 19] by
considering the appropriate $R$ symmetries depending on the number of
supersymmetry. It is known in [19] that the small ${\cal N}=4$ superconformal
algebra [102, 103, 104] is reproduced in [105]. See also the relevant works of
[106, 107]. It is an open problem to observe whether the ${\cal N}=3$
supersymmetric $W_{1+\infty}[\lambda]$ exists or not from the above ${\cal
N}=4$ supersymmetric $W_{1+\infty}[\lambda]$ algebra by truncation or from the
direct construction by using the free field realization (2.1).

So far, we have considered the case of the dimension of three point vertex
$d_{V}=5$ but we can think of the higher derivative cases [74, 71, 108] where
$d_{V}$ can be either $d_{V}=7$ or $d_{V}=9$. It would be interesting to
consider the massive case [109] in the context of [110]. We are not sure
whether the ${\cal N}=1$ supersymmetric Maxwell Einstein theory with scalars
exists or not which combines the previous subsection $5.3.2$ with the
subsection $5.3.3$. It is an open problem to obtain the ${\cal N}=1$
supergravity theory coupled to the most general matter. As mentioned in the
introduction, as a next step, it is an open problem to construct the soft
current algebra for the ${\cal N}=8$ supergravity theory studied in [28, 29].
It is not obvious to observe how we can construct the ${\cal N}=8$
supersymmetric $W_{1+\infty}[\lambda]$ algbra in two dimensions explicitly.

Acknowledgments

CA thanks M. Pate for the discussion on the possibility of nontrivial
anticommutators between the fermions in the context of [74] and on the several
aspects of three point vertices in [10], M. Rocek for the explanation on the
birth of supergravity [69], T. Rahnuma and R.K. Singh for an email
correspondence on [28, 29] and J.-H. Park for the discussion on the presence
of supertranslation in the double field theory. This work was supported by the
National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT) (No. 2023R1A2C1003750).

##  Appendix A  The seven commutators (and OPEs) of section $2$

The seven commutator relations in section $2$ can be summarized as

$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda})_{m}}{(\tilde{W}_{h_{2}}^{\lambda})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3}-2)!$
$\displaystyle\times$
$\displaystyle\bigg{(}\,\,\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-1]_{k}$
$\displaystyle-$
$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}\,(\tilde{W}_{h_{3}}^{\lambda})_{m+n},$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda})_{m}}{(\tilde{W}_{h_{2}}^{\lambda,\hat{A}})_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3}-2)!$
$\displaystyle\times$
$\displaystyle\bigg{(}\,\,\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-1]_{k}$
$\displaystyle-$
$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}\,(\tilde{W}_{h_{3}}^{\lambda,\hat{A}})_{m+n},$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda,\hat{A}})_{m}}{(\tilde{W}_{h_{2}}^{\lambda,\hat{B}})_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{h_{3}=2}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3}-2)!$
$\displaystyle\times$
$\displaystyle\Bigg{[}\bigg{(}\,\,\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-1]_{k}$
$\displaystyle-$
$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}$
$\displaystyle\times$
$\displaystyle\bigg{(}\frac{2}{K}\delta^{\hat{A}\hat{B}}\,\,(\tilde{W}_{h_{3}}^{\lambda})_{m+n}+d^{\hat{A}\hat{B}\hat{C}}\,(\tilde{W}_{h_{3}}^{\lambda,\hat{C}})_{m+n}\bigg{)}$
$\displaystyle+$
$\displaystyle\bigg{(}\,\,\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-1]_{k}$
$\displaystyle+$
$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}$
$\displaystyle\times$ $\displaystyle
i\,f^{\hat{A}\hat{B}\hat{C}}\,(\tilde{W}_{h_{3}}^{\lambda,\hat{C}})_{m+n}\,\Bigg{]},$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\lambda})_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=1}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3})!$
$\displaystyle\times$
$\displaystyle\bigg{(}\,\,\tilde{T}_{R}^{\,\,h_{1},h_{2},h_{3},k}\,\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[r+h_{2}-\tfrac{1}{2}]_{k}$
$\displaystyle-$
$\displaystyle\tilde{T}_{L}^{\,\,h_{1},h_{2},h_{3},k}\,\,[m+h_{1}-1]_{k}\,[r+h_{2}-\tfrac{1}{2}]_{h_{1}+h_{2}-h_{3}-k-1}\bigg{)}\,(Q_{h_{3}+\frac{1}{2}}^{\lambda})_{m+r},$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\lambda,\hat{A}})_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=1}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3})!$
$\displaystyle\times$
$\displaystyle\bigg{(}\,\,\tilde{T}_{R}^{\,\,h_{1},h_{2},h_{3},k}\,\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[r+h_{2}-\tfrac{1}{2}]_{k}$
$\displaystyle-$
$\displaystyle\tilde{T}_{L}^{\,\,h_{1},h_{2},h_{3},k}\,\,[m+h_{1}-1]_{k}\,[r+h_{2}-\tfrac{1}{2}]_{h_{1}+h_{2}-h_{3}-k-1}\bigg{)}\,(Q_{h_{3}+\frac{1}{2}}^{\lambda,\hat{A}})_{m+r},$
$\displaystyle\commutator{(W_{h_{1}}^{\lambda,\hat{A}})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\lambda})_{r}}$
$\displaystyle=$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\lambda,\hat{A}})_{r}},$
$\displaystyle\commutator{(\tilde{W}_{h_{1}}^{\lambda,\hat{A}})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{\lambda,\hat{B}})_{r}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{h_{3}=1}^{h_{1}+h_{2}-1\,\,}\sum_{k=0}^{h_{1}+h_{2}-h_{3}-1}\,(-1)^{h_{3}}\,(4q)^{h_{1}+h_{2}-h_{3}-2}\,(2h_{3})!$
(A.1) $\displaystyle\times$
$\displaystyle\Bigg{[}\bigg{(}\,\,\tilde{T}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-\tfrac{1}{2}]_{k}$
$\displaystyle-$
$\displaystyle\tilde{T}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-\tfrac{1}{2}]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}$
$\displaystyle\times$
$\displaystyle\bigg{(}\frac{2}{K}\delta^{\hat{A}\hat{B}}\,\,(Q_{h_{3}+\frac{1}{2}}^{\lambda})_{m+r}+d^{\hat{A}\hat{B}\hat{C}}\,(Q_{h_{3}+\frac{1}{2}}^{\lambda,\hat{C}})_{m+r}\bigg{)}$
$\displaystyle+$
$\displaystyle\bigg{(}\,\,\tilde{T}^{\,\,h_{1},h_{2},h_{3},k}_{R}\,[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}\,[n+h_{2}-\tfrac{1}{2}]_{k}$
$\displaystyle+$
$\displaystyle\tilde{T}^{\,\,h_{1},h_{2},h_{3},k}_{L}\,[m+h_{1}-1]_{k}\,[n+h_{2}-\tfrac{1}{2}]_{h_{1}+h_{2}-h_{3}-k-1}\,\,\bigg{)}$
$\displaystyle\times$ $\displaystyle
i\,f^{\hat{A}\hat{B}\hat{C}}\,(Q_{h_{3}+\frac{1}{2}}^{\lambda,\hat{C}})_{m+r}\,\Bigg{]},$

where the structure constants are given by

$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{\,R}(\lambda)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=1}^{h_{1}-1}\sum_{i_{2}=1}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r-1)!}\,\beta^{i_{1}-1}(h_{1},\lambda)\beta^{i_{2}-1}(h_{2},\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r-1}{h_{3}-2}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle\tilde{S}^{\,\,h_{1},h_{2},h_{3},k}_{\,L}(\lambda)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=1}^{h_{1}-1}\sum_{i_{2}=1}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r-1)!}\,\beta^{i_{1}-1}(h_{1},\lambda)\beta^{i_{2}-1}(h_{2},\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r-1}{h_{3}-2}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle\tilde{T}_{\,R}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r+1)!}\,\Big{(}2a^{i_{1}}(h_{1},\lambda+\tfrac{1}{2})-\alpha^{i_{1}}(h_{1},\lambda)\Big{)}\beta^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle\tilde{T}_{\,L}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r+1)!}\,\Big{(}2a^{i_{1}}(h_{1},\lambda)-\alpha^{i_{1}}(h_{1},\lambda)\Big{)}\beta^{i_{2}}(h_{2}+1,\lambda)$
(A.2) $\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]}.$

Note that there are some identities between the structure constants in (2.20).
The first and the fourth of Appendix (A.1) appeared in (2.19) and (2.17)
respectively. Of course, the above structure constants become the ones in [13]
for the vanishing $\lambda$.

The corresponding seven OPEs can be obtained by the description in [19] from
the above commutator relations Appendix (A.1) together with Appendix (A.2).

##  Appendix B  The details of the subsection $3.1$

###  B.1  The five (anti)commutators in the subsection $3.1$

For convenience, we present the five (anti)commutators corresponding to
(3.1.2)

$\displaystyle\commutator{(W_{F,h_{1}}^{ab})_{m}}{(W_{F,h_{2}}^{cd})_{n}}$
$\displaystyle=$ $\displaystyle
N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}}\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}\Big{)}q^{2(h_{1}-2)}\,c_{W_{F,h_{1}}}\,[m+h_{1}-1]_{2h_{1}-1}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,p_{F}^{h_{1},h_{2},h}(m,n,0)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{F,h_{1}+h_{2}-h-2}^{bd})_{m+n}+\delta^{ad}(-1)^{h}\,(W_{F,h_{1}+h_{2}-h-2}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{F,h_{1}+h_{2}-h-2}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{F,h_{1}+h_{2}-h-2}^{ac})_{m+n}\Bigg{]},$
$\displaystyle\commutator{(W_{B,h_{1}}^{ab})_{m}}{(W_{B,h_{2}}^{cd})_{n}}$
$\displaystyle=$ $\displaystyle
N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}}\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}\Big{)}q^{2(h_{1}-2)}\,c_{W_{B,h_{1}}}\,[m+h_{1}-1]_{2h_{1}-1}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{h=-1}^{h_{1}+h_{2}-4}q^{h}\,p_{B}^{h_{1},h_{2},h}(m,n,0)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{B,h_{1}+h_{2}-h-2}^{bd})_{m+n}+\delta^{ad}(-1)^{h}\,(W_{B,h_{1}+h_{2}-h-2}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{B,h_{1}+h_{2}-h-2}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}-h-2}^{ac})_{m+n}\Bigg{]},$
$\displaystyle\commutator{(W_{F,h_{1}}^{ab})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{cd})_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,q_{F}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,0)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ad}\,(Q_{h_{1}+h_{2}-h-\frac{3}{2}}^{cb})_{m+r}+\delta^{bd}(-1)^{h_{1}}\,(Q_{F,h_{1}+h_{2}-h-\frac{3}{2}}^{ca})_{m+r}\Bigg{]},$
$\displaystyle\commutator{(W_{B,h_{1}}^{ab})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{cd})_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-3}q^{h}\,q_{B}^{h_{1},h_{2}+\frac{1}{2},h}(m,r,0)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}\,(-1)^{h_{1}}\,(Q_{h_{1}+h_{2}-h-\frac{3}{2}}^{bd})_{m+r}+\delta^{bc}\,(W_{F,h_{1}+h_{2}-h-\frac{3}{2}}^{ad})_{m+r}\Bigg{]},$
$\displaystyle\anticommutator{(Q_{h_{1}+\frac{1}{2}}^{ab})_{r}}{(Q_{h_{1}+\frac{1}{2}}^{cd})_{s}}$
$\displaystyle=$ $\displaystyle
N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}+1}\delta^{ac}\delta^{bd}\Big{)}q^{2(h_{1}-1)}\,c_{Q_{h_{1}}}\,[r+h_{1}-\tfrac{1}{2}]_{2h_{1}}$
(B.1) $\displaystyle-$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-1}q^{h}\,o_{F}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s,0)\,\delta^{ac}\,(-1)^{h_{1}+h}\,(W_{F,h_{1}+h_{2}-h}^{bd})_{r+s}$
$\displaystyle-$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-2}q^{h}\,o_{B}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s,0)\,\delta^{bd}\,(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}-h}^{ac})_{r+s}.$

The generalized hypergeometric function is denoted by

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

and mode dependent function is given by

$\displaystyle N^{h_{1},h_{2}}_{h}(m,n)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\sum_{l=0}^{h+1}(-1)^{l}\left(\begin{array}[]{c}h+1\\\ l\\\
\end{array}\right)[h_{1}-1+m]_{h+1-l}[h_{1}-1-m]_{l}$ (B.7)
$\displaystyle\times\\!$ $\displaystyle[h_{2}-1+n]_{l}[h_{2}-1-n]_{h+1-l}.$
(B.8)

Let us introduce the three kinds of structure constants in [111] together with
Appendix (B.4) and Appendix (B.8)

$\displaystyle\mathrm{BB}^{h_{1},h_{2}}_{r,\,\pm}(m,n;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1},h_{2}}(m,n)\Bigg{[}\phi_{r}^{h_{1},h_{2}}(\mu,1)\pm\phi_{r}^{h_{1},h_{2}}(1-\mu,1)\Bigg{]},$
$\displaystyle\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{r,\,\pm}(m,\rho;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1},h_{2}+\frac{1}{2}}(m,\rho)\Bigg{[}\phi_{r+1}^{h_{1},h_{2}+1}(\mu,\frac{3\pm
1}{2})$ $\displaystyle\pm$
$\displaystyle\phi_{r+1}^{h_{1},h_{2}+1}(1-\mu,\frac{3\pm 1}{2})\Bigg{]},$
$\displaystyle\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{r,\,\pm}(\rho,\omega;\mu)$
$\displaystyle\equiv$
$\displaystyle-\frac{1}{(r-1)!}N_{r-2}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(\rho,\omega)\Bigg{[}\phi_{r+1}^{h_{1}+1,h_{2}+1}(\mu,\frac{3\pm
1}{2})$ (B.9) $\displaystyle\pm$
$\displaystyle\phi_{r+1}^{h_{1}+1,h_{2}+1}(1-\mu,\frac{3\pm 1}{2})\Bigg{]}.$

Then the structure constants in section $3$, from Appendix (B.9), are given by

$\displaystyle p_{F,h}^{h_{1},h_{2}}(m,n,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\Bigg{[}\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,+}+\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle p_{B,h}^{h_{1},h_{2}}(m,n,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\Bigg{[}\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,+}-\mathrm{BB}^{h_{1},h_{2}}_{h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{F,2h}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{(2h_{1}-2h-3)}{16(h+1)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{F,2h+1}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,+}-\frac{(h_{1}-h-2)}{4(2h+3)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{B,2h}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,+}-\frac{(2h_{1}-2h-3)}{16(h+1)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle q_{B,2h+1}^{h_{1},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\frac{1}{8}\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,+}-\frac{(h_{1}-h-2)}{4(2h+3)}\,\mathrm{BF}^{h_{1},h_{2}+\frac{1}{2}}_{2h+3,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{F,2h}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,+}-\frac{2(h_{1}+h_{2}-h)}{(2h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{F,2h+1}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{2(h_{1}+h_{2}-h)-1}{2(h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{B,2h}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,+}+\frac{2(h_{1}+h_{2}-h)}{(2h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+1,\,-}\Bigg{]}_{\mu=2\lambda},$
$\displaystyle o_{B,2h+1}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}(m,n,\lambda)$
$\displaystyle=$
$\displaystyle\Bigg{[}-\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,+}+\frac{2(h_{1}+h_{2}-h)-1}{2(h+1)}\,\mathrm{FF}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{2h+2,\,-}\Bigg{]}_{\mu=2\lambda}.$
(B.10)

At $\lambda=0$, the above structure constants reduce to the ones in [112,
113].

###  B.2  The $q\rightarrow 0$ limit at $\lambda=0$ in the subsection $3.1$

We present some $q$ terms for Appendix (B.1) after taking the rescalings in
(3.7)

$\displaystyle\commutator\Big{(W_{F,h_{1}}^{ab})_{m}}{(W_{F,h_{2}}^{cd})_{n}}$
$\displaystyle=$ $\displaystyle
q^{4(h_{1}-2)}\,N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}}\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}\Big{)}\,c_{W_{F,h_{1}}}\,[m+h_{1}-1]_{2h_{1}-1}$
$\displaystyle+$
$\displaystyle\frac{1}{4q^{2}}\,\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{F,h_{1}+h_{2}-1}^{bd})_{m+n}-\delta^{ad}\,(W_{F,h_{1}+h_{2}-1}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{F,h_{1}+h_{2}-1}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{F,h_{1}+h_{2}-1}^{ac})_{m+n}\Bigg{]}$
$\displaystyle+$ $\displaystyle
q^{0}\,\frac{1}{2}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{F,h_{1}+h_{2}-2}^{bd})_{m+n}+\delta^{ad}\,(W_{F,h_{1}+h_{2}-2}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{F,h_{1}+h_{2}-2}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{F,h_{1}+h_{2}-2}^{ac})_{m+n}\Bigg{]}+\cdots,$
$\displaystyle\commutator\Big{(W_{B,h_{1}}^{ab})_{m}}{(W_{B,h_{2}}^{cd})_{n}}$
$\displaystyle=$ $\displaystyle
q^{4(h_{1}-2)}\,N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}}\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}\Big{)}\,c_{W_{B,h_{1}}}\,[m+h_{1}-1]_{2h_{1}-1}$
$\displaystyle+$
$\displaystyle\frac{1}{4q^{2}}\,\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{B,h_{1}+h_{2}-1}^{bd})_{m+n}-\delta^{ad}\,(W_{B,h_{1}+h_{2}-1}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{B,h_{1}+h_{2}-1}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}-1}^{ac})_{m+n}\Bigg{]}$
$\displaystyle+$ $\displaystyle
q^{0}\,\frac{1}{2}\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,$ $\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}(-1)^{h_{1}}\,(W_{B,h_{1}+h_{2}-2}^{bd})_{m+n}+\delta^{ad}\,(W_{B,h_{1}+h_{2}-2}^{cb})_{m+n}$
$\displaystyle+$
$\displaystyle\delta^{bc}\,(W_{B,h_{1}+h_{2}-2}^{ad})_{m+n}+\delta^{bd}(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}-2}^{ac})_{m+n}\Bigg{]}+\cdots,$
$\displaystyle\commutator{(W_{F,h_{1}}^{ab})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{cd})_{r}}$
$\displaystyle=$
$\displaystyle-\frac{1}{4q^{2}}\,\Bigg{[}\delta^{ad}\,(Q_{h_{1}+h_{2}-\frac{1}{2}}^{cb})_{m+r}+\delta^{bd}(-1)^{h_{1}}\,(Q_{F,h_{1}+h_{2}-\frac{1}{2}}^{ca})_{m+r}\Bigg{]}$
$\displaystyle+$ $\displaystyle
q^{0}\frac{(2h_{2}^{2}-h_{1}-4h_{2}+2h_{1}h_{2}+1)\Big{(}(2h_{2}-1)m-2(h_{1}-1)n\Big{)}}{2(2h_{2}-1)(2h_{1}+2h_{2}-3)}$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ad}\,(Q_{h_{1}+h_{2}-\frac{3}{2}}^{cb})_{m+r}+\delta^{bd}(-1)^{h_{1}}\,(Q_{F,h_{1}+h_{2}-\frac{3}{2}}^{ca})_{m+r}\Bigg{]}+\cdots,$
$\displaystyle\commutator{(W_{B,h_{1}}^{ab})_{m}}{(Q_{h_{2}+\frac{1}{2}}^{cd})_{r}}$
$\displaystyle=$
$\displaystyle\frac{1}{4q^{2}}\,\Bigg{[}\delta^{ac}\,(-1)^{h_{1}}\,(Q_{h_{1}+h_{2}-h-\frac{1}{2}}^{bd})_{m+r}+\delta^{bc}\,(Q_{h_{1}+h_{2}-h-\frac{1}{2}}^{ad})_{m+r}\Bigg{]}$
$\displaystyle+$ $\displaystyle
q^{0}\frac{(2h_{2}^{2}-h_{1}-4h_{2}+2h_{1}h_{2}+2)\Big{(}(2h_{2}-1)m-2(h_{1}-1)n\Big{)}}{2(2h_{2}-1)(2h_{1}+2h_{2}-3)}$
$\displaystyle\times$
$\displaystyle\Bigg{[}\delta^{ac}\,(-1)^{h_{1}}\,(Q_{h_{1}+h_{2}-h-\frac{3}{2}}^{bd})_{m+r}+\delta^{bc}\,(Q_{h_{1}+h_{2}-h-\frac{3}{2}}^{ad})_{m+r}\Bigg{]}+\cdots,$
$\displaystyle\anticommutator{(Q_{h_{1}+\frac{1}{2}}^{ab})_{r}}{(Q_{h_{2}+\frac{1}{2}}^{cd})_{s}}$
$\displaystyle=$ $\displaystyle
q^{4(h_{1}-1)}\,N\,\delta^{h_{1}h_{2}}\Big{(}(-1)^{h_{1}+1}\delta^{ac}\delta^{bd}\Big{)}\,c_{Q_{h_{1}}}\,[r+h_{1}-\tfrac{1}{2}]_{2h_{1}}$
(B.11) $\displaystyle-$ $\displaystyle
q^{0}\,2\,\delta^{ac}\,(-1)^{h_{1}}\,(W_{F,h_{1}+h_{2}}^{bd})_{r+s}$
$\displaystyle-$ $\displaystyle
q^{0}\,2\,\delta^{bd}\,(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}}^{ac})_{r+s}\quad$
$\displaystyle-$ $\displaystyle
q^{2}\,\frac{4(2h_{1}h_{2}-h_{1}-h_{2}+1)\Big{(}(2h_{2}-1)r-(2h_{1}-1)s\Big{)})}{(2h_{1}-1)(2h_{2}-1)}$
$\displaystyle\times$
$\displaystyle\delta^{ac}\,(-1)^{h_{1}}\,(W_{F,h_{1}+h_{2}-1}^{bd})_{r+s}$
$\displaystyle-$ $\displaystyle
q^{2}\,\frac{4(2h_{1}h_{2}-h_{1}-h_{2})\Big{(}(2h_{2}-1)r-(2h_{1}-1)s\Big{)})}{(2h_{1}-1)(2h_{2}-1)}$
$\displaystyle\times$
$\displaystyle\delta^{bd}\,(-1)^{h_{2}}\,(W_{B,h_{1}+h_{2}-1}^{ac})_{r+s}$
$\displaystyle+$ $\displaystyle\cdots,$

where the abbreviated parts in Appendix (B.11) are the higher orders of $q$.
We can easily remove the divergent $\frac{1}{q^{2}}$ terms by linear
combinations between the currents.

##  Appendix C  The (anti)commutators at $\lambda=\frac{1}{4}$ in the
subsection $3.2$

We present the (anti)commutators as follows:

$\displaystyle\commutator{\Big{(}V^{(h_{1}),+}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}V^{(h_{2}),+}_{\frac{1}{4}}\Big{)}_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})+S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+n}$
$\displaystyle+$
$\displaystyle\sum_{h_{3}=1,\,\text{odd}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}2\,(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})-S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+n},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),-}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}V^{(h_{2}),-}_{\frac{1}{4}}\Big{)}_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}\frac{1}{4}(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})+S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+n}$
$\displaystyle+$
$\displaystyle\sum_{h_{3}=1,\,\text{odd}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}\frac{1}{2}\,(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})-S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+n},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),+}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}V^{(h_{2}),-}_{\frac{1}{4}}\Big{)}_{n}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}\frac{1}{2}(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}-S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})+S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+n}$
$\displaystyle+$
$\displaystyle\sum_{h_{3}=1,\,\text{odd}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}(2h_{3}-1)!\,$
$\displaystyle\times$
$\displaystyle\Big{(}-S_{F,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})-S_{B,R}^{\,\,h_{1},h_{2},h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[n+h_{2}-1]_{k}\,\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+n},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),+}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}Q^{(h_{2}),+}_{\frac{1}{4}}\Big{)}_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}(2h_{3}-2)!\,$
$\displaystyle\times$
$\displaystyle\Bigg{[}\,\Big{(}-T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+r}$
$\displaystyle+$
$\displaystyle\Big{(}T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+r}\Bigg{]},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),-}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}Q^{(h_{2}),+}_{\frac{1}{4}}\Big{)}_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}\frac{1}{2}\,(2h_{3}-2)!\,$
$\displaystyle\times$
$\displaystyle\Bigg{[}\,\Big{(}T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+r}$
$\displaystyle+$
$\displaystyle\Big{(}-T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+r}\Bigg{]},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),+}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}Q^{(h_{2}),-}_{\frac{1}{4}}\Big{)}_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}(2h_{3}-2)!\,$
$\displaystyle\times$
$\displaystyle\Bigg{[}\,\Big{(}T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+r}$
$\displaystyle+$
$\displaystyle\Big{(}-T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+r}\Bigg{]},$
$\displaystyle\commutator{\Big{(}V^{(h_{1}),-}_{\frac{1}{4}}\Big{)}_{m}}{\Big{(}Q^{(h_{2}),-}_{\frac{1}{4}}\Big{)}_{r}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{even}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-1}\frac{1}{2}(2h_{3}-2)!\,$
$\displaystyle\times$
$\displaystyle\Bigg{[}\,\Big{(}-T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{m+r}$
$\displaystyle+$
$\displaystyle\Big{(}T_{F}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})+\bar{T}_{B}^{\,\,h_{1},h_{2}-1,h_{3}-1,k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[m+h_{1}-1]_{h_{1}+h_{2}-h_{3}-k-1}[r+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}Q^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{m+r}\Bigg{]},$
$\displaystyle\anticommutator{\Big{(}Q^{(h_{1}),\pm}_{\frac{1}{4}}\Big{)}_{r}}{\Big{(}Q^{(h_{2}),\pm}_{\frac{1}{4}}\Big{)}_{s}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{Even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-2}(2h_{3}-1)!$
$\displaystyle\times$ $\displaystyle\Bigg{[}\,\Big{(}\pm
U_{F}^{\,\,h_{1}-1,h_{2}-1,h_{3},k}(\tfrac{1}{4})\pm
U_{B}^{\,\,h_{2}-1,h_{1}-1,h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[r+h_{1}-\tfrac{3}{2}]_{h_{1}+h_{2}-h_{3}-k-2}[s+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{r+s}\Bigg{]}$
$\displaystyle+$
$\displaystyle\sum_{h_{3}=1,\,\text{odd}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-2}2\,(2h_{3}-1)!$
$\displaystyle\times$ $\displaystyle\Bigg{[}\,\Big{(}\pm
U_{F}^{\,\,h_{1}-1,h_{2}-1,h_{3},k}(\tfrac{1}{4})\mp
U_{B}^{\,\,h_{2}-1,h_{1}-1,h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[r+h_{1}-\tfrac{3}{2}]_{h_{1}+h_{2}-h_{3}-k-2}[s+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{r+s}\Bigg{]},$
$\displaystyle\anticommutator{\Big{(}Q^{(h_{1}),\pm}_{\frac{1}{4}}\Big{)}_{r}}{\Big{(}Q^{(h_{2}),\mp}_{\frac{1}{4}}\Big{)}_{s}}$
$\displaystyle=$
$\displaystyle\sum_{h_{3}=2,\,\text{Even}}^{h_{1}+h_{2}-2}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-2}(2h_{3}-1)!$
(C.1) $\displaystyle\times$ $\displaystyle\Bigg{[}\,\Big{(}\pm
U_{F}^{\,\,h_{1}-1,h_{2}-1,h_{3},k}(\tfrac{1}{4})\mp
U_{B}^{\,\,h_{2}-1,h_{1}-1,h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[r+h_{1}-\tfrac{3}{2}]_{h_{1}+h_{2}-h_{3}-k-2}[s+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}V^{(h_{3}),+}_{\frac{1}{4}}\Big{)}_{r+s}\Bigg{]}$
$\displaystyle+$
$\displaystyle\sum_{h_{3}=1,\,\text{odd}}^{h_{1}+h_{2}-1}\sum_{k=0}^{\,\,h_{1}+h_{2}-h_{3}-2}2\,(2h_{3}-1)!$
$\displaystyle\times$ $\displaystyle\Bigg{[}\,\Big{(}\pm
U_{F}^{\,\,h_{1}-1,h_{2}-1,h_{3},k}(\tfrac{1}{4})\pm
U_{B}^{\,\,h_{2}-1,h_{1}-1,h_{3},k}(\tfrac{1}{4})\Big{)}$
$\displaystyle\times$
$\displaystyle[r+h_{1}-\tfrac{3}{2}]_{h_{1}+h_{2}-h_{3}-k-2}[s+h_{2}-\tfrac{3}{2}]_{k}\,\Big{(}V^{(h_{3}),-}_{\frac{1}{4}}\Big{)}_{r+s}\Bigg{]},$

where the structure constants are, from [19],

$\displaystyle S^{\,\,h_{1},h_{2},h_{3},k}_{F,\,R}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda+\tfrac{1}{2})a^{i_{2}}(h_{2},\lambda+\tfrac{1}{2})$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle S^{\,\,h_{1},h_{2},h_{3},k}_{F,\,L}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda+\tfrac{1}{2})a^{i_{2}}(h_{2},\lambda+\tfrac{1}{2})$
$\displaystyle m$ $\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle S^{\,\,h_{1},h_{2},h_{3},k}_{B,\,R}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda\big{)}a^{i_{2}}(h_{2},\lambda\big{)}$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle S^{\,\,h_{1},h_{2},h_{3},k}_{B,\,L}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda)a^{i_{2}}(h_{2},\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle T_{F}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r+1)!}\,a^{i_{1}}(h_{1},\lambda+\tfrac{1}{2})\beta^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle\bar{T}_{F}^{\,\,h_{1},h_{2},h_{3},k}(h_{1},\lambda)$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda+\tfrac{1}{2})\alpha^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}-1}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle T_{B}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}-1}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r+1)!}\,a^{i_{1}}(h_{1},\lambda)\beta^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle\bar{T}_{B}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}}\,\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,a^{i_{1}}(h_{1},\lambda)\alpha^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{-h_{3}+r}{-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}-1}(1-r-2\lambda+j)\Bigg{]},$
$\displaystyle U_{F}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}}\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,\beta^{i_{1}}(h_{1}+1,\lambda)\alpha^{i_{2}}(h_{2}+1,\lambda)$
$\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{2}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{-h_{2}+i_{2}+k}\prod_{j=0}^{r-h_{3}}(-r-2\lambda+j)\Bigg{]},$
$\displaystyle U_{B}^{\,\,h_{1},h_{2},h_{3},k}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{i_{1}=0}^{h_{1}-1}\sum_{i_{2}=0}^{h_{2}}\sum_{r=0}^{i_{1}+i_{2}}\Bigg{[}\frac{(-1)^{i_{1}+i_{2}}}{(h_{3}+r)!}\,\beta^{i_{1}}(h_{1}+1,\lambda)\alpha^{i_{2}}(h_{2}+1,\lambda)$
(C.2) $\displaystyle\times$
$\displaystyle\binom{r}{h_{3}-1}\binom{i_{1}}{i_{1}+i_{2}-r}\binom{1-h_{3}+r}{1-h_{1}+i_{1}+k}\prod_{j=0}^{r-h_{3}}(1-r-2\lambda+j)\Bigg{]}.$

Note that on the right hand sides of Appendix (C.1), the dummy variables
$h_{3}$ are restricted to be the even for the modes of bosonic currents having
$+$ superscript and the odd for the modes of bosonic currents having $-$
superscript. They are restricted to be the even for the modes of fermionic
currents having $\pm$ superscript.

In [22, 23], there exists other subalgebra generated by

$\displaystyle V^{(h),+}_{\lambda},h=2,4,6,\cdots,\qquad
V^{(h),-}_{\lambda},h=1,3,5,\cdots,$ $\displaystyle
Q^{(h),+}_{\lambda},h=1,3,5,\cdots,\qquad Q^{(h),-}_{\lambda},h=1,3,5,\cdots,$
(C.3)

which is the nonsupersymmetric case. We can express the corresponding
(anti)commutators similarly. The first three commutators and the last two
anticommutators of Appendix (C.1) remain the same. Of course, the fermions are
restricted to have odd number of weights as in Appendix (C.3). For the next
four commutators of Appendix (C.1), we need to write down for the odd number
for the fermions as in Appendix (C.3) with the extra minus signs on the right
hand sides. In other words, the structure constants are opposite, compared to
the supersymmetric case in Appendix (C.1).

##  Appendix D  The $q\rightarrow 0$ limit at general $\lambda$ in the
subsection $3.4$

We present some $q$ terms for Appendix $D$ of [19] after a rescaling (3.14) as
follows:

$\displaystyle\commutator{(W^{\lambda}_{F,\,h_{1}})_{m}}{(W^{\lambda}_{F,\,h_{2}})_{n}}=q^{2(h_{1}+h_{2}-4)}\,K\,N\,c^{h_{1},h_{2}}_{W_{F}}(\lambda)\,[m+h_{1}-1]_{h_{1}+h_{2}-1}$
$\displaystyle+q^{0}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W^{\lambda}_{F,\,h_{1}+h_{2}-2})_{m+n}+\cdots\,,$
$\displaystyle\commutator{(W^{\lambda}_{F,\,h_{1}})_{m}}{(W^{\lambda,\hat{A}}_{F,\,h_{2}})_{n}}=q^{0}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W^{\lambda,\hat{A}}_{F,\,h_{1}+h_{2}-2})_{m+n}+\cdots\,,$
$\displaystyle\commutator{(W^{\lambda,\hat{A}}_{F,\,h_{1}})_{m}}{(W^{\lambda,\hat{B}}_{F,\,h_{2}})_{n}}=q^{2(h_{1}+h_{2}-2)}\,N\,\delta^{\hat{A}\hat{B}}\,c^{h_{1},h_{2}}_{W_{F}}(\lambda)\,[m+h_{1}-1]_{h_{1}+h_{2}-1}$
$\displaystyle+q^{4}\,\delta^{\hat{A}\hat{B}}\,\frac{1}{K}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W^{\lambda}_{F,\,h_{1}+h_{2}-2})_{m+n}$
$\displaystyle+q^{2}\,\frac{1}{2}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{F,\,h_{1}+h_{2}-2})_{m+n}$
$\displaystyle-q^{0}\,\frac{i}{4}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\lambda,\hat{C}}_{F,\,h_{1}+h_{2}-1})_{m+n}+\cdots\,,$
$\displaystyle\commutator{(W^{\lambda}_{B,\,h_{1}})_{m}}{(W^{\lambda}_{B,\,h_{2}})_{n}}=q^{2(h_{1}+h_{2}-4)}\,K\,N\,c^{h_{1},h_{2}}_{W_{B}}(\lambda)\,[m+h_{1}-1]_{h_{1}+h_{2}-1}$
$\displaystyle+q^{0}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W^{\lambda}_{B,\,h_{1}+h_{2}-2})_{m+n}+\cdots\,,$
$\displaystyle\commutator{(W^{\lambda}_{B,\,h_{1}})_{m}}{(W^{\lambda,\hat{A}}_{B,\,h_{2}})_{n}}=q^{0}\,\Big{(}(h_{2}-1)m-(h_{1}-1)n\Big{)}\,(W^{\lambda,\hat{A}}_{B,\,h_{1}+h_{2}-2})_{m+n}+\cdots\,,$