A Deformed Supersymmetric $w_{1+\infty}$ Symmetry

in the Celestial Conformal Field Theory

Changhyun Ahn

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

<EMAIL_ADDRESS>
Abstract

By using the $K$-free complex bosons and the $K$-free complex fermions, we
construct the ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra which is
the matrix generalization of previous ${\cal N}=2$ supersymmetric $W_{\infty}$
algebra. By twisting this ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$
algebra, we obtain the ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra
which is the matrix generalization of known ${\cal N}=1$ supersymmetric
topological $W_{\infty}$ algebra. From this two-dimensional symmetry algebra,
we propose the operator product expansion (OPE) between the soft graviton and
gravitino (as a first example), at nonzero deformation parameter, in the
supersymmetric Einstein-Yang-Mills theory explicitly. Other six OPEs between
the graviton, gravitino, gluon and gluino can be determined completely. At
vanishing deformation parameter, we reproduce the known result of Fotopoulos,
Stieberger, Taylor and Zhu on the above OPEs via celestial holography.

###### Contents

  1. 1  Introduction
  2. 2  The ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry
    1. 2.1 The supersymmetric $W_{\infty}^{K,L}$ algebra with $U(K)\times U(L)$ symmetry: Review
    2. 2.2 Free field realization: Review
    3. 2.3  The ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry
      1. 2.3.1  The $W_{1+\infty}^{K}$ algebra
      2. 2.3.2 The $W_{\infty}^{K}$ algebra
      3. 2.3.3 The commutators between the bosonic and fermionic currents
      4. 2.3.4 The commutators between the other bosonic and fermionic currents
    4. 2.4 Free field realization
    5. 2.5  The existence of ${\cal N}=2$ supersymmetric $w_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry?
  3. 3  The ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$ symmetry
    1. 3.1  The ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$ symmetry
      1. 3.1.1  The commutators between the bosonic currents
      2. 3.1.2  The commutators between the bosonic currents and the fermionic currents
    2. 3.2 Free field realization
    3. 3.3  The ${\cal N}=1$ supersymmetric $w_{\infty}^{K}$ algebra with $U(K)$ symmetry
    4. 3.4 The seven OPEs
    5. 3.5 The possible realization in the ${\cal N}=1$ supersymmetric Einstein-Yang-Mills theory
      1. 3.5.1 The OPE between the soft positive helicity graviton and the soft positive helicity gravitino
      2. 3.5.2 The OPE between the soft positive helicity gravitons
  4. 4  Conclusions and outlook
  5. A  The remaining (anti)commutator relations in the ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry
    1. A.1 The commutators between the bosonic and the other fermionic currents
    2. A.2 The commutators between the other bosonic and the other fermionic currents
    3. A.3 The anticommutators between the fermionic currents
  6. B  The operator product expansions in the ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$ symmetry
    1. B.1  The seven OPEs for fixed $h_{1}$ and $h_{2}$
    2. B.2 The structure constants for fixed $h_{1}$, $h_{2}$
  7. C  Other OPEs for soft currents in the supersymmetric Einstein-Yang-Mills theory
    1. C.1 The OPE between the graviton and the gluino
    2. C.2 The OPE between the gluon and the gravitino
    3. C.3 The OPE between the gluon and the gluino
    4. C.4  The OPE between the graviton and the gluon
    5. C.5 The OPE between the gluons

##  1  Introduction

Recently, the celestial holography has been proposed by the fact that there
exists a duality between the gravitational scattering in asymptotically flat
spacetimes and the conformal field theory which lives on the celestial sphere.
See the review papers [1, 2, 3, 4] on the celestial holography. By using the
low energy scattering problems, the symmetry algebra of the conformal field
theory for flat space (a celestial conformal field theory) has been studied in
[5]. Moreover, in [6], the group of symmetries on the celestial sphere
satisfies the wedge subalgebra of $w_{1+\infty}$ algebra [7]. This implies
that we should understand the structures behind these findings thoroughly in
order to check the above duality. In [8], the supersymmetric $w_{1+\infty}$
algebra has been identified with the corresponding soft current algebra in the
supersymmetric Einstein-Yang-Mills theory. The relevant works on the celestial
holography in the various directions can be found in [9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35]. See [1] for more complete literatures.

The higher-spin extension of the Virasoro algebra has been found by
Zamolodchikov [36]. The so-called $W_{3}$ algebra consists of the spin-$2$
stress energy tensor and the spin-$3$ current. Subsequently, this $W_{3}$
algebra is generalized to the $W_{N}$ algebra [37, 38] which is generated by
the spin-$2$ stress energy tensor and the higher-spin currents of each spin,
$s=3,4,\cdots,N$. It is also possible to construct a linear $W_{\infty}$
algebra [39, 40] generated by the currents with spins $s=2,3,4,\cdots,\infty$.
A simple contraction of the $W_{\infty}$ algebra leads to the $w_{\infty}$
algebra [7]. Moreover, the $W_{1+\infty}$ algebra [41] contains all spins
$s=1,2,3,\cdots,\infty$. The ${\cal N}=2$ supersymmetric $W_{\infty}$ algebra
[42] whose bosonic sector is given by $W_{\infty}$ and $W_{1+\infty}$ is
obtained.

Bakas and Kiritsis [43] have found the $W_{\infty}^{K}$ algebra which is an
$U(K)$-matrix generalization of $W_{\infty}$ algebra. For each current of spin
$s$, there are $K^{2}$ multicomponent generators. Odake and Sano [44] also
have constructed the $W_{1+\infty}^{L}$ algebra which is an $U(L)$-matrix
extension of $W_{1+\infty}$ algebra. There exist $L^{2}$ multicomponent
generators for each current of spin $s$. Furthermore, the supersymmetric
$W_{\infty}^{K,L}$ algebra, whose bosonic sector is given by $W_{\infty}^{K}$
and $W_{1+\infty}^{L}$, has been studied in [45].

In this paper, by taking the condition $K=L$, we construct the ${\cal N}=2$
supersymmetric $W_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry. Due
to the above condition $K=L$, we can multiply the generators in the fermionic
currents. The ${\cal N}=2$ supersymmetry is reduced to the ${\cal N}=1$
supersymmetry by topological twisting [46, 47, 48]. Then the ${\cal N}=1$
supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$ symmetry is obtained. That
is, we obtain the matrix generalization of [48]. The seven commutator
relations between the bosonic and fermionic currents can be written in terms
of the various structure constants and the deformation parameter. By
considering the vanishing limit of this deformation parameter, we reproduce
the previous result of [49]. We propose that the OPEs between the graviton,
gravitino, gluon and gluino in the supersymmetric Einstein-Yang-Mills theory
can be determined from the above two-dimensional symmetry algebra 111 We are
focusing on the soft currents where the celestial operators have the specific
conformal dimensions for the bosonic and fermionic fields. For the former
$\Delta=1,0,-1,\cdots$ and for the latter
$\Delta=\frac{1}{2},-\frac{1}{2},-\frac{3}{2},\cdots$. See also [5, 31]..

For the nonvanishing deformation parameter, the commutators contain the
possible terms in the right hand sides. In general, the structure constants
depend on the two modes of the commutator and the weights. Among the weights,
the weights $h_{1}$ and $h_{2}$ appearing in the left hand side of the
commutator are given. On the other hand, the weight $h$ appearing in the right
hand side vary from its lowest value to the highest value depending on the
previous weights $h_{1}$ and $h_{2}$. The weight $h$ plays the role of dummy
variable in the summation of the right hand side of the commutator. We would
like to determine the OPEs between the above soft currents in the
supersymmetric Einstein-Yang-Mills theory by looking at the two dimensional
symmetry algebra characterized by seven commutators. Then the question is how
we can determine the OPEs in the soft currents which will eventually lead to
the commutators we have found in two dimensional conformal field theory after
performing the appropriate contour integrals.

As mentioned before, the structure constants consists of mode dependent part
and mode independent part and they do depend on the above three weights
dependence. We should figure out how the mode dependent part can be read off
from the relevant OPE between the soft currents because the mode independent
part can be multiplied into this inside of the dummy variable weight $h$. From
the experience of the various contour integrals [5], we expect that there
should be $h$ dependence in the OPE when we consider the case of the nonzero
deformation parameter. Once we have obtained the correct OPEs which produces
the mode dependent part of the commutators, then it is straightforward to
determine the full OPEs by multiplying the weights dependence parts and
summing over above dummy variable $h$ within the possible range.

In section $2$, we obtain the ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$
algebra after reviewing the supersymmetric $W_{\infty}^{K,L}$ algebra. In
section $3$, we determine the ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$
algebra. The free field realization is given. At the vanishing deformation
parameter, the previous result [8] is reproduced. vWe also present the seven
commutator relations in terms of the corresponding OPEs. Finally, we propose
its realization in the ${\cal N}=1$ supersymmetric Einstein-Yang-Mills theory.
In section $4$, we summarize what we have obtained in this paper. In
Appendices, we provide some details in sections $2$ and $3$.

##  2  The ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra with
$U(K)\times U(K)$ symmetry

###  2.1 The supersymmetric $W_{\infty}^{K,L}$ algebra with $U(K)\times U(L)$
symmetry: Review

The nontrivial (anti)commutator relations of $W_{\infty}^{K,L}$ algebra [45]
(See also [50]) are given by

$\displaystyle\big{[}(W^{\bar{\alpha}{\beta}}_{\mathrm{F},h_{1}})_{m},(W^{\bar{\gamma}{\delta}}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-3}\,\frac{\lambda^{h}}{2}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\,\Bigg{[}\delta^{\bar{\gamma}{\beta}}\,(W^{\bar{\alpha}{\delta}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle(-1)^{h}\delta^{\bar{\alpha}{\delta}}(W^{\bar{\gamma}{\beta}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\Bigg{]}+c_{W_{\mathrm{F},h_{1}}}(m)\delta^{\bar{\alpha}{\delta}}\delta^{{\beta}\bar{\gamma}}\delta^{h_{1}h_{2}}\lambda^{2(h_{1}-2)}\delta_{m+n}\
,$
$\displaystyle\big{[}(W^{\bar{a}{b}}_{\mathrm{B},h_{1}})_{m},(W^{\bar{c}{d}}_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-4}\,\frac{\lambda^{h}}{2}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\,\Bigg{[}\delta^{\bar{c}{b}}\,(W^{\bar{a}{d}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle(-1)^{h}\delta^{\bar{a}{d}}(W^{\bar{c}{b}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}\Bigg{]}+c_{W_{\mathrm{B},h_{1}}}(m)\delta^{\bar{a}{d}}\delta^{{b}\bar{c}}\delta^{h_{1}h_{2}}\lambda^{2(h_{1}-2)}\delta_{m+n}\
,$
$\displaystyle\big{[}(W^{\bar{\alpha}{\beta}}_{\mathrm{F},h_{1}})_{m},(Q^{\bar{a}{\gamma}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\delta^{\bar{\alpha}{\gamma}}\,(Q^{\bar{a}{\beta}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W^{\bar{\alpha}{\beta}}_{\mathrm{F},h_{1}})_{m},(\bar{Q}^{{a}\bar{\gamma}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,(-1)^{h}\,q_{\mathrm{F}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\delta^{\beta\bar{\gamma}}\,(\bar{Q}^{a\bar{\alpha}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W^{\bar{a}{b}}_{\mathrm{B},h_{1}})_{m},(Q^{\bar{c}{\alpha}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\delta^{\bar{c}{b}}\,(Q^{\bar{a}{\alpha}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W^{\bar{a}{b}}_{\mathrm{B},h_{1}})_{m},(\bar{Q}^{c\bar{\alpha}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,(-1)^{h}\,q_{\mathrm{B}}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\delta^{\bar{a}c}\,(\bar{Q}^{{b}\bar{\alpha}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\\{(Q^{\bar{a}{\alpha}}_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}^{{b}\bar{\beta}}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum_{h=0}^{h_{1}+h_{2}-1}\,\lambda^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\delta^{\bar{a}{b}}\,(W^{\bar{\beta}\alpha}_{\mathrm{F},h_{1}+h_{2}-h})_{r+s}$
(2.1) $\displaystyle+$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-2}\,\lambda^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\delta^{\alpha\bar{\beta}}\,(W^{\bar{a}{b}}_{\mathrm{B},h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$ $\displaystyle
c_{Q\bar{Q}_{h_{1}+\frac{1}{2}}}(r)\,\delta^{\bar{a}{b}}\,\delta^{{\alpha}\bar{\beta}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}+\frac{1}{2}-1)}\,\delta_{r+s}\
.$

The bosonic $W_{\infty}^{K}$ subalgebra corresponding to the second equation
of (2.1) is generated by the $U(K)$-adjoint $W^{\bar{a}{b}}_{\mathrm{B},h}$
with an integer weight $h=2,3,\cdots,\infty$. The subscript $B$ stands for the
bilinear of complex free bosons in next subsection. The fundamental index
$a,b,\cdots$ of $U(K)$ runs over $a,b,\cdots=1,2,\cdots,K$ and the
antifundamental index $\bar{a},\bar{b},\cdots$ of $U(K)$ runs over
$\bar{a},\bar{b},\cdots=1,2,\cdots,K$. The bosonic $W_{1+\infty}^{L}$
subalgebra corresponding to the first equation of (2.1) is generated by the
$U(L)$-adjoint $W^{\bar{\alpha}{\beta}}_{\mathrm{F},h}$ with an integer weight
$h=1,2,\cdots,\infty$. Note the presence of weight-$1$ current. The subscript
$F$ stands for the bilinear of complex free fermions in next subsection. The
fundamental index $\alpha,\beta,\cdots$ of $U(L)$ runs over
$\alpha,\beta,\cdots=1,2,\cdots,L$ and the antifundamental index
$\bar{\alpha},\bar{\beta},\cdots$ of $U(L)$ runs over
$\bar{\alpha},\bar{\beta},\cdots=1,2,\cdots,L$. There are also the
bifundamental $Q^{\bar{a}{\alpha}}_{h+\frac{1}{2}}$ and the bifundamental
$\bar{Q}^{{b}\bar{\beta}}_{h+\frac{1}{2}}$ under the $U(K)\times U(L)$ with
the half-integer weight $h+\frac{1}{2}=\frac{3}{2},\frac{5}{2},\cdots$ for the
remaining (anti)commutator relations. Note that the lower and upper limits for
the dummy variable $h$ in (2.1) can be determined by the fact that i) the
maximum weight for the current in the right hand side is equal to the sum of
two weights in the left hand side minus one and ii) the minimum weight for the
current in the right hand side is equal to $2,1$ or $\frac{3}{2}$ as above.

The $\lambda$ is a deformation parameter 222 This corresponds to the parameter
$q$ in [45] and is nothing to do with the one in the higher spin algebra in
[50]. and the central terms in (2.1) except the $\lambda$-dependent factors
are given by

$\displaystyle c_{W_{\mathrm{F},h}}(m)$ $\displaystyle=$ $\displaystyle
N\,k\,\frac{2^{2(h-3)}\,(h-1)!)\,(h-1)!}{(2h-3)!!\,(2h-1)!!}\,\prod_{j=1-h}^{h-1}(m+j)\
,$ $\displaystyle c_{W_{\mathrm{B},h}}(m)$ $\displaystyle=$ $\displaystyle
N\,k\,\frac{2^{2(h-3)}(h-2)!\,h!}{(2h-3)!!\,(2h-1)!!}\,\prod_{j=1-h}^{h-1}(m+j)\
,$ $\displaystyle c_{Q\bar{Q}_{h}}(r)\\!$ $\displaystyle=\\!$ $\displaystyle
N\,k\,\frac{2^{2(h-\frac{3}{2})}(h-\frac{3}{2})!\,(h-\frac{1}{2})!}{(2h-2)!!\,(2h-2)!!}\,\prod_{j=\frac{1}{2}-h}^{h-\frac{3}{2}}(r+j+\frac{1}{2})\,.$
(2.2)

There exists an overall factor $N$ which is related to the number of free
complex bosons (or fermions). The $k$ is the level of $\hat{SU}(L)$ and the
corresponding weight-one current is given by
$\frac{4\,\lambda}{\sqrt{L\,N}}\,W^{\bar{\alpha}{\beta}}_{\mathrm{F},1}\,\delta_{\beta\bar{\alpha}}$
with $c_{W_{\mathrm{F},1}}(m)=\frac{N\,m}{16}$. By introducing the $k$ copies
of the free field realization, we construct the general level $k$ realization
[45] because $W_{\infty}^{K,L}$ algebra is linear. The Virasoro central charge
is given by $c=Nk(2K+L)$ from
$c_{W_{\mathrm{F},2}}(m)=\frac{1}{12}\,N\,k\,m(m^{2}-1)$ and
$c_{W_{\mathrm{B},2}}(m)=\frac{1}{6}\,N\,k\,m(m^{2}-1)$ from (2.2) and the
Sugawara stress energy tensor is given by
$W^{\bar{a}{b}}_{\mathrm{B},2}\,\delta_{b\bar{a}}-W^{\bar{\alpha}{\beta}}_{\mathrm{F},2}\,\delta_{\beta\bar{\alpha}}$.

The mode-dependent structure constants appearing in (2.1) are described as
follows:

$\displaystyle p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{1}{2(h+1)!}\,\phi^{h_{1},h_{2}}_{h}(0,\textstyle{-\frac{1}{2}})\,N^{h_{1},h_{2}}_{h}(m,n)\
,$ $\displaystyle p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{1}{2(h+1)!}\,\phi^{h_{1},h_{2}}_{h}(0,0)\,N^{h_{1},h_{2}}_{h}(m,n)\
,$ $\displaystyle q_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{(-1)^{h}}{4(h+2)!}\Bigg{[}(h_{1}-1)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,0)$
$\displaystyle-\\!$
$\displaystyle(h_{1}-h-3)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,\textstyle{-\frac{1}{2}})\Bigg{]}\,N^{h_{1},h_{2}}_{h}(m,r)\
,$ $\displaystyle q_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{-1}{4(h+2)!}\Bigg{[}(h_{1}-h-2)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,0)-(h_{1})\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,\textstyle{-\frac{1}{2}})\Bigg{]}\,N^{h_{1},h_{2}}_{h}(m,r)\
,$ $\displaystyle o_{\mathrm{F}}^{h_{1},h_{2},h}(r,s)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{4(-1)^{h}}{h!}\Bigg{[}(h_{1}+h_{2}-1-h)\,\phi^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{h}(\textstyle{\frac{1}{2}},\textstyle{-\frac{1}{4}})$
$\displaystyle-\\!$
$\displaystyle(h_{1}+h_{2}-\frac{3}{2}-h)\,\phi^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{h+1}(\textstyle{\frac{1}{2}},\textstyle{-\frac{1}{4}})\Bigg{]}\,N^{h_{1},h_{2}}_{h-1}(r,s)\
,$ $\displaystyle o_{\mathrm{B}}^{h_{1},h_{2},h}(r,s)\\!$
$\displaystyle\equiv\\!$
$\displaystyle-\frac{4}{h!}\Bigg{[}(h_{1}+h_{2}-2-h)\,\phi^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{h}(\textstyle{\frac{1}{2}},\textstyle{-\frac{1}{4}})$
(2.3) $\displaystyle-\\!$
$\displaystyle(h_{1}+h_{2}-\frac{3}{2}-h)\,\phi^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2}}_{h+1}(\textstyle{\frac{1}{2}},\textstyle{-\frac{1}{4}})\Bigg{]}\,N^{h_{1},h_{2}}_{h-1}(r,s)\
.$

The structure constants are polynomials in the modes. The modes $m,n,\cdots$
are integers and the modes $r,s,\cdots$ are half-integers. We introduce the
following quantities

$\displaystyle N^{h_{1},h_{2}}_{h}(m,n)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\sum_{l=0}^{h+1}(-1)^{l}\left(\begin{array}[]{c}h+1\\\ l\\\
\end{array}\right)[h_{1}-1+m]_{h+1-l}[h_{1}-1-m]_{l}$ $\displaystyle\times\\!$
$\displaystyle[h_{2}-1+n]_{l}[h_{2}-1-n]_{h+1-l}\ ,$
$\displaystyle\phi^{h_{1},h_{2}}_{h}(x,y)\\!$ $\displaystyle\equiv\\!$
$\displaystyle{}_{4}F_{3}\Bigg{[}\begin{array}[]{c}-\frac{1}{2}-x-2y,\frac{3}{2}-x+2y,-\frac{h+1}{2}+x,-\frac{h}{2}+x\\\
-h_{1}+\frac{3}{2},-h_{2}+\frac{3}{2},h_{1}+h_{2}-h-\frac{3}{2}\end{array};1\Bigg{]}\
.$ (2.9)

We use the falling Pochhammer symbol $[a]_{n}\equiv a(a-1)\cdots(a-n+1)$ in
(2.9) and we use the binomial coefficients for parentheses. Moreover, the
generalized hypergeometric function, with four upper arguments $a_{i}$, three
lower arguments $b_{i}$ and variable $z$, is defined as the series

$\displaystyle{}_{4}F_{3}\Bigg{[}\begin{array}[]{c}a_{1},a_{2},a_{3},a_{4}\\\
b_{1},b_{2},b_{3}\end{array};z\Bigg{]}=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}(a_{2})_{n}(a_{3})_{n}(a_{4})_{n}}{(b_{1})_{n}(b_{2})_{n}(b_{3})_{n}}\frac{z^{n}}{n!}\,,$
(2.12)

where the rising Pochhammer symbol $(a)_{n}\equiv a(a+1)\cdots(a+n-1)$ is used
in (2.12) 333 In the right hand side of (2.9), the $h_{1}$ and $h_{2}$ play
the role of $(i+2)$ (or $(i+\frac{3}{2})$) and $(j+2)$ (or $(j+\frac{3}{2})$)
of [45]. The $h$ corresponds to their $r$. Due to the typos in [42] (See also
the footnote $2$ of [48]), the structure constants in [45] are different from
the ones in [42]. For example, the structure constant $a^{i\,\alpha}_{l}(m,r)$
of [42] is given by our $(-1)^{h}\,q_{B}^{h_{1},h_{2},h}(m,r)$ with the
identification $h_{1}=i+2,h_{2}=\alpha+\frac{3}{2},h=l-1$ and the structure
constant $\tilde{a}^{i\,\alpha}_{l}(m,r)$ of [42] is given by our
$(-1)^{h}\,q_{F}^{h_{1},h_{2},h}(m,r)$ with the identification
$h_{1}=i+2,h_{2}=\alpha+\frac{3}{2},h=l-1$..

In this paper, by acting the generators of $U(K)$ (or $U(L)$) with the
contractions of the (anti)fundamental indices on the (anti)commutator
relations in (2.1), we will determine the ${\cal N}=2$ supersymmetric
$W_{\infty}^{K,K}$ algebra and the ${\cal N}=1$ supersymmetric
$W_{\infty}^{K}$ algebra. Then the fermionic currents will not have any
(anti)fundamental indices.

###  2.2 Free field realization: Review

The $W_{\infty}^{K,L}$ algebra with level $k=1$ is realized by $K$-free
complex bosons of weight-$1$ ($\bar{\partial}\,\phi^{\bar{i},a}$ and
$\bar{\partial}\,\bar{\phi}^{i,\bar{a}}$) and $L$-free complex fermions of
weight-$\frac{1}{2}$ ($\psi^{\bar{i},\alpha}$ and
$\bar{\psi}^{i,\bar{\alpha}}$). The index $i$ is the fundamental index of
$U(N)$ and the index $\bar{i}$ is the antifundamental index of $U(N)$. Their
operator product expansions in the antiholomorphic sector are

$\displaystyle\bar{\partial}\,\bar{\phi}^{i,\bar{a}}(\bar{z})\,\bar{\partial}\,\phi^{\bar{j},b}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\frac{1}{(\bar{z}-\bar{w})^{2}}\,\delta^{i\bar{j}}\,\delta^{\bar{a}b}+\cdots\
,$
$\displaystyle\bar{\psi}^{i,\bar{\alpha}}(\bar{z})\,\psi^{\bar{j},\beta}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\frac{1}{(\bar{z}-\bar{w})}\,\delta^{i\bar{j}}\,\delta^{\bar{\alpha}\beta}+\cdots\
.$ (2.13)

The $U(N)$-singlet currents of $W_{\infty}^{K,L}$ algebra are described by the
bilinears of these free fields as follows (See also [51]):

$\displaystyle W^{\bar{\alpha}\beta}_{\mathrm{F},h}$ $\displaystyle=$
$\displaystyle
n_{W_{\mathrm{F},h}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\
\end{array}\right)^{2}\,(\,\bar{\partial}^{h-l-1}\bar{\psi}^{i,\bar{\alpha}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\beta}\,)\,,$
(2.16) $\displaystyle W^{\bar{a}b}_{\mathrm{B},h}$ $\displaystyle=$
$\displaystyle
n_{W_{\mathrm{B},h}}\sum_{l=0}^{h-2}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}\frac{(-1)^{l}}{(h-1)}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h-1\\\ l+1\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l-1}\bar{\phi}^{i,\bar{a}}\,\bar{\partial}^{l+1}\phi^{\bar{\imath},b}\,)\,,$
(2.21) $\displaystyle Q^{\bar{a}\alpha}_{h+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle
n_{Q_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\bar{\phi}^{i,\bar{a}}\bar{\partial}^{l}\psi^{\bar{\imath},\alpha}\,)\,,$
(2.26) $\displaystyle\bar{Q}^{a\bar{\alpha}}_{h+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle
n_{\bar{Q}_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{h-1+l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\phi^{\bar{\imath},a}\bar{\partial}^{l}\bar{\psi}^{i,\bar{\alpha}}\,)\,.$
(2.31)

The $l=0$ cases of single summations correspond to the lowest weights,
$1,2,\frac{3}{2}$ and $\frac{3}{2}$. The normalizations are given by

$\displaystyle n_{W_{\mathrm{F},h}}\\!$ $\displaystyle=\\!$
$\displaystyle\frac{2^{h-3}(h-1)!}{(2h-3)!!}\,\lambda^{h-2}\ ,\qquad
n_{W_{\mathrm{B},h}}=\frac{2^{h-3}\,h!}{(2h-3)!!}\,\lambda^{h-2}\ ,$
$\displaystyle n_{Q_{h+\frac{1}{2}}}\\!$ $\displaystyle=\\!$
$\displaystyle\frac{2^{h-\frac{1}{2}}h!}{(2h-1)!!}\,\lambda^{h-1}=n_{\bar{Q}_{h+\frac{1}{2}}}\
.$ (2.32)

Then we can check that the modes of (2.31) satisfy the previous
(anti)commutator relations (2.1) by using the mode expansion for the normal
ordering between the free fields in the conformal field theory. After using
the (anti)commutator relations corresponding to (2.13), the left hand sides of
(2.1) contain the quadratic free fields having the explicit modes (where the
coefficients depend on the weights, the modes and the dummy variable from the
infinite sum) and the central terms. Similarly, the right hand sides of (2.1)
contain the quadratic free fields and the central terms in the presence of the
nontrivial structure constants (2.3). For several low values of the weights
and the modes, we can check the several nontrivial identities. Alternatively,
after using the Thielemans package [52] for low values of the weights,
simplifying the right hand sides of the OPEs between the currents and
rewriting down them in terms of the (anti)commutator relations with the help
of the explicit formula in [53, 54], the previous algebra (2.1) can be checked
also explicitly 444This paragraph is based on the discussion with S. Odake
some years ago..

###  2.3  The ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra with
$U(K)\times U(K)$ symmetry

Let us consider the case of $K=L$. Then the number of (anti)fundamental
indices is the same. From the decomposition of $U(K+K)=U(K)\oplus
U(K)\oplus({\bf K},\bar{\bf K})\oplus(\bar{\bf K},{\bf K})$, the generators
consists of $t^{\hat{A}}_{\alpha\bar{\beta}}$, $t^{\hat{A}}_{a\bar{b}}$,
$t^{\hat{A}}_{\alpha\bar{a}}$ and $t^{\hat{A}}_{a\bar{\alpha}}$ in addition to
$\delta_{\alpha\bar{\beta}}$, $\delta_{a\bar{b}}$, $\delta_{\alpha\bar{a}}$
and $\delta_{a\bar{\alpha}}$ with $\hat{A}=1,2,\cdots,(K^{2}-1)$.

By multiplying the generators into the four kinds of currents, we obtain four
kinds of singlets and adjoints of $U(K)$ as follows:

$\displaystyle W_{\mathrm{F},h}\\!$ $\displaystyle\equiv\\!$ $\displaystyle
W^{\bar{\alpha}\beta}_{\mathrm{F},h}\,\delta_{\beta\bar{\alpha}}\ ,\qquad
W^{\hat{A}}_{\mathrm{F},h}\equiv
W^{\bar{\alpha}\beta}_{\mathrm{F},h}\,t^{\hat{A}}_{\beta\bar{\alpha}}\ ,$
$\displaystyle W_{\mathrm{B},h}\\!$ $\displaystyle\equiv\\!$ $\displaystyle
W^{\bar{a}b}_{\mathrm{B},h}\,\delta_{b\bar{a}}\ ,\qquad
W^{\hat{A}}_{\mathrm{B},h}\equiv
W^{\bar{a}b}_{\mathrm{B},h}\,t^{\hat{A}}_{b\bar{a}}\ ,$ $\displaystyle
Q_{h+\frac{1}{2}}\\!$ $\displaystyle\equiv\\!$ $\displaystyle
Q^{\bar{a}\alpha}_{h+\frac{1}{2}}\,\delta_{\alpha\bar{a}}\ ,\qquad
Q^{\hat{A}}_{h+\frac{1}{2}}\equiv
Q^{\bar{a}\alpha}_{h+\frac{1}{2}}\,t^{\hat{A}}_{\alpha\bar{a}}\ ,$
$\displaystyle\bar{Q}_{h+\frac{1}{2}}\\!$ $\displaystyle\equiv\\!$
$\displaystyle\bar{Q}^{a\bar{\alpha}}_{h+\frac{1}{2}}\,\delta_{a\bar{\alpha}}\
,\qquad\bar{Q}^{\hat{A}}_{h+\frac{1}{2}}\equiv\bar{Q}^{a\bar{\alpha}}_{h+\frac{1}{2}}\,t^{\hat{A}}_{a\bar{\alpha}}\
,\qquad\hat{A}=1,2,\cdots,(K^{2}-1)\ .$ (2.33)

Now we would like to rewrite down (2.1) in terms of (2.33) after multiplying
the various generators 555From now on, we do not have to distinguish the two
(anti) fundamental indices..

####  2.3.1  The $W_{1+\infty}^{K}$ algebra

Now we can multiply the generators into the first equation of (2.1) and the
three commutator relations can be obtained as follows:

$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(W_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\,(W_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$ $\displaystyle
K\,c_{W_{\mathrm{F},h_{1}}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}\,\delta_{m+n}\
,$
$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(W^{\hat{A}}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\,(W^{\hat{A}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(W^{\hat{B}}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\sum^{h_{1}+h_{2}-3}_{h=-1,\mbox{\footnotesize
odd}}\,\lambda^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
(2.34) $\displaystyle+\\!$ $\displaystyle
c_{W_{\mathrm{F},h_{1}}}\,\delta^{\hat{A}\hat{B}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}\,\delta_{m+n}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{F}}^{h_{1},h_{2},h}(m,n)\,\Big{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(W_{\mathrm{F},h_{1}+h_{2}-2-h})_{m+n}\Big{]}\
.$

This $W_{1+\infty}^{K}$ algebra (or $\hat{SU}(K)_{k}$ $W_{1+\infty}$ algebra)
was found in [44]. The first equation of (2.34) generated by the singlet
current of $U(K)$ is $W_{1+\infty}$ algebra and its extension with the adjoint
of $U(K)$ appears in the remaining equations. In the last equation of (2.34),
the identity for the product of two generators
$t^{\hat{A}}\,t^{\hat{B}}=\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,{\bf
1}_{K}+\frac{1}{2}\,(i\,f+d)^{\hat{A}\hat{B}\hat{C}}\,t^{\hat{C}}$ is used.
Note that the weights in the right hand side appear in even or odd integers.
Of course, by taking the contractions of the currents with the vanishing
$\lambda$ limit, the first equation reduces to the $w_{1+\infty}$ algebra
(which is the weight-$1$ extension of $w_{\infty}$ algebra [7]) as shown in
[8] 666 By redefining the currents of weights $2,3,4$ nonlinearly, we can
decouple the weight-$1$ current from other currents. That is, there are no
singular OPEs between the weight-$1$ current and others. The OPEs between the
above currents of weights $2,3,4$ do not contain the weight-$1$ current at the
poles in the right hand side. The lowest pole we are considering contains the
weight-$4$ current. For the poles having the higher spin current of spins,
$5,6,\cdots$, we need to find out the corresponding redefined currents step by
step. We expect that this will be true for higher weights. See also the
relevant paper [55]..

####  2.3.2 The $W_{\infty}^{K}$ algebra

The second equation of (2.1) can be rewritten as

$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(W_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-4}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\,(W_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$ $\displaystyle
K\,c_{W_{\mathrm{B},h_{1}}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}\,\delta_{m+n}\
,$
$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(W^{\hat{A}}_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-4}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\,(W^{\hat{A}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(W^{\hat{B}}_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\sum^{h_{1}+h_{2}-4}_{h=-1,\mbox{\footnotesize
odd}}\,\lambda^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}$
(2.35) $\displaystyle+\\!$ $\displaystyle
c_{W_{\mathrm{B},h_{1}}}\,\delta^{\hat{A}\hat{B}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}\,\delta_{m+n}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-4}_{h=0,\mbox{\footnotesize
even}}\,\lambda^{h}\,p_{\mathrm{B}}^{h_{1},h_{2},h}(m,n)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(W_{\mathrm{B},h_{1}+h_{2}-2-h})_{m+n}\Bigg{]}\
.$

This $W_{\infty}^{K}$ algebra was found in [43]. The first equation of (2.35)
generated by the singlet current of $U(K)$ is $W_{\infty}$ algebra and its
extension with the adjoint of $U(K)$ appears in the remaining equations. The
algebraic structure of (2.35) looks similar to the one of (2.34). The upper
bound of dummy variable $h$ and the structure constants are different from
each other. By taking the contractions of the currents with the vanishing
$\lambda$ limit, the first equation reduces to the $w_{\infty}$ algebra [7].

####  2.3.3 The commutators between the bosonic and fermionic currents

In this case, we have four commutator relations after multiplying the
generators into the third equation of (2.1)

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

####  2.3.4 The commutators between the other bosonic and fermionic currents

From the fifth equation of (2.1), the following four commutator relations can
be obtained by multiplying the generators

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

We present here (2.36) and (2.37) which are necessary to describe the
discussion of next section and the remaining (anti)commutator relations are
presented in Appendix $A$.

###  2.4 Free field realization

From (2.33) and (2.31) with (2.32), we obtain the following free field
realization

$\displaystyle W_{\mathrm{F},h}$ $\displaystyle=$ $\displaystyle
n_{W_{\mathrm{F},h}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\
\end{array}\right)^{2}\,(\,\bar{\partial}^{h-l-1}\bar{\psi}^{i,\bar{\alpha}}\,\delta_{\beta\bar{\alpha}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\beta}\,)\,,$
(2.40) $\displaystyle W^{\hat{A}}_{\mathrm{F},h}$ $\displaystyle=$
$\displaystyle
n_{W_{\mathrm{F},h}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\
\end{array}\right)^{2}\,(\,\bar{\partial}^{h-l-1}\bar{\psi}^{i,\bar{\alpha}}\,t^{\hat{A}}_{\beta\bar{\alpha}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\beta}\,)\,,$
(2.43) $\displaystyle W_{\mathrm{B},h}$ $\displaystyle=$ $\displaystyle
n_{W_{\mathrm{B},h}}\sum_{l=0}^{h-2}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}\frac{(-1)^{l}}{(h-1)}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h-1\\\ l+1\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l-1}\bar{\phi}^{i,\bar{a}}\,\delta_{b\bar{a}}\,\bar{\partial}^{l+1}\phi^{\bar{\imath},b}\,)\,,$
(2.48) $\displaystyle W^{\hat{A}}_{\mathrm{B},h}$ $\displaystyle=$
$\displaystyle
n_{W_{\mathrm{B},h}}\sum_{l=0}^{h-2}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}\frac{(-1)^{l}}{(h-1)}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h-1\\\ l+1\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l-1}\bar{\phi}^{i,\bar{a}}\,t^{\hat{A}}_{b\bar{a}}\,\bar{\partial}^{l+1}\phi^{\bar{\imath},b}\,)\,,$
(2.53) $\displaystyle Q_{h+\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
n_{Q_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\bar{\phi}^{i,\bar{a}}\,\delta_{\alpha\bar{a}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\alpha}\,)\,,$
(2.58) $\displaystyle Q^{\hat{A}}_{h+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle
n_{Q_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\bar{\phi}^{i,\bar{a}}\,t^{\hat{A}}_{\alpha\bar{a}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\alpha}\,)\,,$
(2.63) $\displaystyle\bar{Q}_{h+\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
n_{\bar{Q}_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{h-1+l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\phi^{\bar{\imath},a}\,\delta_{a\bar{\alpha}}\,\bar{\partial}^{l}\bar{\psi}^{i,\bar{\alpha}}\,)\,,$
(2.68) $\displaystyle\bar{Q}^{\hat{A}}_{h+\frac{1}{2}}$ $\displaystyle=$
$\displaystyle
n_{\bar{Q}_{h+\frac{1}{2}}}\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{h-1+l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\phi^{\bar{\imath},a}\,t^{\hat{A}}_{a\bar{\alpha}}\,\bar{\partial}^{l}\bar{\psi}^{i,\bar{\alpha}}\,)\,.$
(2.73)

The number $N$ which is contracted in (2.73) does not play any role of our
discussion. In this basis, the singlet and adjoint property of $U(K)$ is
clear. It is evident that the above free field realizations satisfy (2.36) and
(2.37) and Appendices (A.1), (A.3) and (A.5). Except the last two of (2.73),
the remaining ones will be used in next section.

###  2.5  The existence of ${\cal N}=2$ supersymmetric $w_{\infty}^{K,K}$
algebra with $U(K)\times U(K)$ symmetry?

By taking the simple rescalings

$\displaystyle W_{h}\\!$ $\displaystyle\rightarrow\\!$
$\displaystyle\lambda\,W_{h}\ ,\qquad
W_{h}^{\hat{A}}\rightarrow\lambda\,W_{h}^{\hat{A}}\ ,$ $\displaystyle
Q_{h+\frac{1}{2}}\\!$ $\displaystyle\rightarrow\\!$
$\displaystyle\lambda\,Q_{h+\frac{1}{2}}\ ,\qquad
Q^{\hat{A}}_{h+\frac{1}{2}}\rightarrow\lambda\,Q^{\hat{A}}_{h+\frac{1}{2}}\ ,$
(2.74)

and putting the $\lambda$ to vanish, we obtain the following commutator
relations

$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(W^{\hat{B}}_{\mathrm{F},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\,\frac{i}{4}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{F},h_{1}+h_{2}-1})_{m+n}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(W^{\hat{B}}_{\mathrm{B},h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\,\frac{i}{4}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{\mathrm{B},h_{1}+h_{2}-1})_{m+n}\
,$
$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(Q_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\ ,$
$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{i}{8}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}-\frac{1}{4}\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(Q_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\Bigg{]}\
,$
$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(Q_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\ ,$
$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{i}{8}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}+\frac{1}{4}\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}$
(2.75) $\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(Q_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\Bigg{]}\
.$

There are no mode dependent terms in the right hand side. In the OPE language,
the $\frac{1}{\lambda}$ term in the first order pole in the original OPEs
survives. Other reduced commutator relations similar to (2.75) appear in
Appendix $A$. Once we keep the commutators in the bosonic singlet currents,
then we have some $\frac{1}{\lambda}$ dependence in other commutators.

##  3  The ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$
symmetry

###  3.1  The ${\cal N}=1$ supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$
symmetry

####  3.1.1  The commutators between the bosonic currents

In [48], the bosonic current of weight $h$ is given by the linear combination
of $W_{B,h}$, $W_{F,h}$, $\bar{\partial}\,W_{B,h-1}$ and
$\bar{\partial}\,W_{F,h-1}$. In terms of their modes with correct deformation
parameter $\lambda$ (the power of $\lambda$ should be equal to $(h-2)$), we
obtain the following $U(K)$-singlet and $U(K)$-adjoint currents together with
(2.33)

$\displaystyle(W_{h})_{m}$ $\displaystyle\equiv$
$\displaystyle(W_{\mathrm{B},h})_{m}+(W_{\mathrm{F},h})_{m}+\lambda\,\frac{2(h-2)(m+(h-2)+1)}{2(h-2)+1}\,(W_{\mathrm{B},h-1})_{m}$
$\displaystyle-\\!$
$\displaystyle\lambda\,\frac{(2(h-2)+2)(m+(h-2)+1)}{2(h-2)+1}\,(W_{\mathrm{F},h-1})_{m}\
,$ $\displaystyle(W^{\hat{A}}_{h})_{m}$ $\displaystyle\equiv$
$\displaystyle(W^{\hat{A}}_{\mathrm{B},h})_{m}+(W^{\hat{A}}_{\mathrm{F},h})_{m}+\lambda\,\frac{2(h-2)(m+(h-2)+1)}{2(h-2)+1}\,(W^{\hat{A}}_{\mathrm{B},h-1})_{m}$
(3.1) $\displaystyle-\\!$
$\displaystyle\lambda\,\frac{(2(h-2)+2)(m+(h-2)+1)}{2(h-2)+1}\,(W^{\hat{A}}_{\mathrm{F},h-1})_{m}\
.$

According to the construction of (3.1), the lowest value for the weight $h$ of
the bosonic current $W_{B,h}$ is given by $h=2$. When $h=2$, the third term of
each expression vanishes due to the coefficient. The $U(K)$-adjoint current is
new and is the matrix generalization of [48]. The mode $m$ dependence in the
coefficients appears in the third and fourth terms when we write down the mode
of derivative of current in terms of mode of the current itself 777In
principle [55], we can add the weight-$1$ currents by reversing the procedure
in the footnote 6 and obtain the $W_{1+\infty}^{K}$ algebra and by
contractions the corresponding $w_{1+\infty}^{K}$ algebra can be obtained)..

Then we can determine their commutator relations explicitly by using both
(2.34) and (2.35) as follows:

$\displaystyle\big{[}(W_{h_{1}})_{m},(W_{h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-4}_{h=0}\,\lambda^{h}\,q^{h_{1},h_{2},h}(m,n)\,(W_{h_{1}+h_{2}-2-h})_{m+n}+c_{W,h_{1}}(m)\
,$ $\displaystyle\big{[}(W_{h_{1}})_{m},(W^{\hat{A}}_{h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-4}_{h=0}\,\lambda^{h}\,q^{h_{1},h_{2},h}(m,n)\,(W^{\hat{A}}_{h_{1}+h_{2}-2-h})_{m+n}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(W^{\hat{B}}_{h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\sum^{h_{1}+h_{2}-4}_{h=-1}\,\lambda^{h}\,\tilde{q}^{h_{1},h_{2},h}(m,n)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{h_{1}+h_{2}-2-h})_{m+n}$
(3.2) $\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,c_{W,h_{1}}(m)$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-4}_{h=0}\,\lambda^{h}\,q^{h_{1},h_{2},h}(m,n)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{h_{1}+h_{2}-2-h})_{m+n}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(W_{h_{1}+h_{2}-2-h})_{m+n}\Bigg{]}\
.$

Again, the last two relations in (3.2) are new. Compared to (2.35), the
algebraic structure looks similar, but the structure constants are different
from each other and the range for the dummy variable $h$ in the right hand
sides is different. It is claimed in [48] that the algebra in the first
relation of (3.2) is isomorphic to the $W_{\infty}$ algebra [39, 40] 888 It is
known in [48] that the diagonal $W_{\infty}$ algebra from the current $W_{h}$
is generated in $W_{\infty}$ algebra (generated by $W_{B,h}$) and
$W_{1+\infty}$ algebra (generated by $W_{F,h}$). . Here the central terms
appearing in the first and the last equations of (3.2) are given by the
following expression

$\displaystyle c_{W,h_{1}}(m)\\!$ $\displaystyle=\\!$ $\displaystyle
K\Bigg{[}c_{W_{B,h_{1}}}(m)\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}$ (3.3)
$\displaystyle+\\!$
$\displaystyle\lambda\,\frac{2(h_{2}-2)(n+(h_{2}-2)+1)}{2(h_{2}-2)+1}\,c_{W_{B,h_{1}}}(m)\,\delta^{h_{1},h_{2}-1}\,\lambda^{2(h_{1}-2)}\,$
$\displaystyle+\\!$
$\displaystyle\lambda\,\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,c_{W_{B,h_{1}-1}}(m)\,\delta^{h_{1}-1,h_{2}}\,\lambda^{2(h_{1}-1-2)}\,$
$\displaystyle+\\!$
$\displaystyle\lambda^{2}\,\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,\frac{2(h_{2}-2)(n+(h_{2}-2)+1)}{2(h_{2}-2)+1}\,$
$\displaystyle\times\\!$ $\displaystyle
c_{W_{B,h_{1}-1}}(m)\,\delta^{h_{1}-1,h_{2}-1}\,\lambda^{2(h_{1}-1-2)}\,$
$\displaystyle+\\!$ $\displaystyle
c_{W_{F,h_{1}}}(m)\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}-2)}\,$
$\displaystyle-\\!$
$\displaystyle\lambda\,\frac{(2(h_{2}-2)+2)(n+(h_{2}-2)+1)}{2(h_{2}-2)+1}\,c_{W_{F,h_{1}}}(m)\,\delta^{h_{1},h_{2}-1}\,\lambda^{2(h_{1}-2)}\,$
$\displaystyle-\\!$
$\displaystyle\lambda\,\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,c_{W_{F,h_{1}-1}}(m)\,\delta^{h_{1}-1,h_{2}}\,\lambda^{2(h_{1}-1-2)}\,$
$\displaystyle+\\!$
$\displaystyle\lambda^{2}\,\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,\frac{(2(h_{2}-2)+2)(n+(h_{2}-2)+1)}{2(h_{2}-2)+1}\,$
$\displaystyle\times\\!$ $\displaystyle
c_{W_{F,h_{1}-1}}(m)\,\delta^{h_{1}-1,h_{2}-1}\,\lambda^{2(h_{1}-1-2)}\,\Bigg{]}\,\delta_{m+n}\
.$

We can check that the above expression (3.3) vanishes (topological property)
by using the Kronecker delta conditions properly. The second and the sixth,
the third and the seventh, and the remaining ones can be combined as the
independent terms. We introduce the following structure constants

$\displaystyle q^{h_{1},h_{2},h}(m,n)\\!$ $\displaystyle\equiv\\!$
$\displaystyle q_{\mathrm{B}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle+\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{B}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})$
$\displaystyle+\\!$ $\displaystyle
q_{\mathrm{F}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{F}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})\
,$ $\displaystyle\tilde{q}^{h_{1},h_{2},h}(m,n)$ $\displaystyle\equiv$
$\displaystyle q_{\mathrm{B}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
(3.4) $\displaystyle+\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{B}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})$
$\displaystyle-\\!$ $\displaystyle
q_{\mathrm{F}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle+\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{F}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})\
.$

The first relation of (3.4) was found in [48] with our convention and is more
natural in the commutator relation between the bosonic current and the
fermionic current in next subsection. That is the reason why there are shifts
in the weight $h_{2}$ and the mode $n$. The second and the fourth terms have
the explicit mode $m$ dependence due to the derivative terms as we explained
before. Note that there are precise relations between the structure constants
$(p_{B},p_{F})$ which appear in the equations (2.34) and (2.35) and the
structure constants $(q_{F},q_{B})$ which appear in the equations (2.36) and
(2.37) at each term in the commutator relations. Their relations will appear
later.

####  3.1.2  The commutators between the bosonic currents and the fermionic
currents

Now we obtain the commutator relations including the fermionic currents. By
using (2.36), (2.37) and (2.33), the following relations satisfy

$\displaystyle\big{[}(W_{h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,q^{h_{1},h_{2}+1,h}(m,r-\frac{1}{2})\,(Q_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W_{h_{1}})_{m},(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,q^{h_{1},h_{2}+1,h}(m,r-\frac{1}{2})\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,q^{h_{1},h_{2}+1,h}(m,r-\frac{1}{2})\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,\hat{q}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
(3.5) $\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,\check{q}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(Q_{h_{1}+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{]}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,\dot{q}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}-1+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-3}_{h=-1}\,\lambda^{h}\,\bar{q}^{h_{1},h_{2}+\frac{1}{2},h}(m,r)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}-1+h_{2}-\frac{3}{2}-h})_{m+r}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(Q_{h_{1}-1+h_{2}-\frac{3}{2}-h})_{m+r}\Bigg{]}\
.$

Again, the first relation of (3.5) was found in [48]. The remaining ones are
the matrix generalization. In the last equation of (3.5), the following
structure constants are introduced by collecting each contribution

$\displaystyle\hat{q}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle-
q_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)+q_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\ ,$
$\displaystyle\check{q}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle
q_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)+q_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\ ,$
$\displaystyle\dot{q}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle-\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{B}}^{h_{1}-1,h_{2},h}(m,r)$
$\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{F}}^{h_{1}-1,h_{2},h}(m,r)\
,$ $\displaystyle\bar{q}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{B}}^{h_{1}-1,h_{2},h}(m,r)$
(3.6) $\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{\mathrm{F}}^{h_{1}-1,h_{2},h}(m,r)\
.$

Although we introduce the weight $h_{2}$ in the structure constant in (3.6),
their appearance in (3.5) takes the form of $(h_{2}+\frac{1}{2})$ which is the
weight of the fermionic current in the left hand side. The shift in the weight
$h_{1}$ of the structure constants $(q_{B},q_{F})$ in the last two of (3.6)
can be understood from the derivative of the bosonic current in the left hand
side. Moreover, the last three terms in the last equation of (3.5) have the
weight of the first three terms minus one. We can combine the last three terms
by introducing a new dummy variable $(h+1)$ with the first three terms in
addition to other term. In this way, we can simplify the last equation of
(3.5) further.

Then one of our main results with a deformation parameter $\lambda$ is
summarized by (3.2) and (3.5) together with (3.4) and (3.6). We will present
the realization of this algebra in the supersymmetric Einstein-Yang-Mills
theory.

###  3.2 Free field realization

By combining (2.33) and (2.73), we can write down the free field realization
for the singlet current and the adjoint current of $U(K)$ as follows:

$\displaystyle W_{h}\\!$ $\displaystyle=\\!$ $\displaystyle
W_{B,h}+W_{F,h}-\lambda\,\frac{2(h-2)}{2(h-2)+1}\,\bar{\partial}\,W_{B,h-1}+\lambda\,\frac{2(h-2)+2}{2(h-2)+1}\,\bar{\partial}\,W_{F,h-1}\,,$
$\displaystyle=\\!$
$\displaystyle\frac{2^{h-2}(h-1)!}{(2(h-2)+1)!!}\,\lambda^{h-2}\,\sum_{l=0}^{h-2}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}\,(-1)^{l}\,\left(\begin{array}[]{c}h-2\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h-1\\\ l\\\
\end{array}\right)\,$ $\displaystyle\times\\!$
$\displaystyle\Bigg{(}\,\bar{\partial}^{h-l-1}\bar{\phi}^{i,\bar{a}}\,\delta_{b\bar{a}}\,\bar{\partial}^{l+1}\phi^{\bar{\imath},b}+\bar{\partial}^{h-l-1}\bar{\psi}^{i,\bar{\alpha}}\,\delta_{\beta\bar{\alpha}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\beta}\,\Bigg{)}\,,$
$\displaystyle W^{\hat{A}}_{h}\\!$ $\displaystyle=\\!$ $\displaystyle
W^{\hat{A}}_{B,h}+W^{\hat{A}}_{F,h}-\lambda\,\frac{2(h-2)}{2(h-2)+1}\,\bar{\partial}\,W^{\hat{A}}_{B,h-1}+\lambda\,\frac{2(h-2)+2}{2(h-2)+1}\,\bar{\partial}\,W^{\hat{A}}_{F,h-1}\,$
(3.16) $\displaystyle=\\!$
$\displaystyle\frac{2^{h-2}(h-1)!}{(2(h-2)+1)!!}\,\lambda^{h-2}\,\sum_{l=0}^{h-2}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}\,(-1)^{l}\,\left(\begin{array}[]{c}h-2\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h-1\\\ l\\\
\end{array}\right)\,$ $\displaystyle\times\\!$
$\displaystyle\Bigg{(}\,\bar{\partial}^{h-l-1}\bar{\phi}^{i,\bar{a}}\,t^{\hat{A}}_{b\bar{a}}\,\bar{\partial}^{l+1}\phi^{\bar{\imath},b}+\bar{\partial}^{h-l-1}\bar{\psi}^{i,\bar{\alpha}}\,t^{\hat{A}}_{\beta\bar{\alpha}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\beta}\,\Bigg{)}\,.$
(3.17)

There is some difference in the sign when we compare with the result of [48]
because this comes from the fact that we are using different normalization in
the footnote 3.

The remaining superpartner currents come from (2.73) as follows:

$\displaystyle Q_{h+\frac{1}{2}}\\!$ $\displaystyle=\\!$
$\displaystyle\frac{2^{h-\frac{1}{2}}h!}{(2h-1)!!}\,\lambda^{h-1}\,\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\bar{\phi}^{i,\bar{a}}\,\delta_{\alpha\bar{a}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\alpha}\,)\,,$
(3.22) $\displaystyle Q^{\hat{A}}_{h+\frac{1}{2}}\\!$ $\displaystyle=\\!$
$\displaystyle\frac{2^{h-\frac{1}{2}}h!}{(2h-1)!!}\,\lambda^{h-1}\,\sum_{l=0}^{h-1}\sum_{i,\,\bar{\imath}=1}^{N}\delta_{i,\bar{\imath}}(-1)^{l}\left(\begin{array}[]{c}h-1\\\
l\\\ \end{array}\right)\left(\begin{array}[]{c}h\\\ l\\\
\end{array}\right)\,(\,\bar{\partial}^{h-l}\bar{\phi}^{i,\bar{a}}\,t^{\hat{A}}_{\alpha\bar{a}}\,\bar{\partial}^{l}\psi^{\bar{\imath},\alpha}\,)\,.$
(3.27)

In principle, we can check the previous commutator relations by using these
free field realizations, calculating the various OPEs and rewriting down these
in terms of commutator relations. The power of deformation parameter is given
by the weight minus $2$ or $\frac{3}{2}$. As in the abstract, the results of
(3.17) and (3.27) are the matrix generalization of previous work of [48].

###  3.3  The ${\cal N}=1$ supersymmetric $w_{\infty}^{K}$ algebra with $U(K)$
symmetry

In order to keep the nonzero lowest power of the deformation parameter in each
term of (3.2) and (3.5), we consider the scales for the currents with the
deformation parameter whose power depends on the weights. According to the
following transformations,

$\displaystyle W_{h}\\!$ $\displaystyle\rightarrow\\!$
$\displaystyle\lambda^{h-2}\,W_{h}\ ,\qquad
W_{h}^{\hat{A}}\rightarrow\lambda^{h}\,W_{h}^{\hat{A}}\ ,$ $\displaystyle
Q_{h+\frac{1}{2}}\\!$ $\displaystyle\rightarrow\\!$
$\displaystyle\lambda^{h+\frac{1}{2}-2}\,Q_{h+\frac{1}{2}}\ ,\qquad
Q^{\hat{A}}_{h+\frac{1}{2}}\rightarrow\lambda^{h+\frac{1}{2}}\,Q^{\hat{A}}_{h+\frac{1}{2}}\
,$ (3.28)

we obtain the following the ${\cal N}=1$ supersymmetric $w_{1+\infty}^{K}$
algebra with $U(K)$ symmetry after redefining as (3.28) and taking
$\lambda\rightarrow 0$ limit with the help of (3.4) and (3.6)

$\displaystyle\big{[}(W_{h_{1}})_{m},(W_{h_{2}})_{n}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!q^{h_{1},h_{2},0}(m,n)\,(W_{h_{1}+h_{2}-2})_{m+n}$
$\displaystyle=\\!$
$\displaystyle\Big{[}m(h_{2}-1)-n(h_{1}-1)\Big{]}\,(W_{h_{1}+h_{2}-2})_{m+n}\
,$ $\displaystyle\big{[}(W_{h_{1}})_{m},(W^{\hat{A}}_{h_{2}})_{n}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!q^{h_{1},h_{2},0}(m,n)\,(W^{\hat{A}}_{h_{1}+h_{2}-2})_{m+n}$
$\displaystyle=\\!$
$\displaystyle\Big{[}m(h_{2}-1)-n(h_{1}-1)\Big{]}\,(W^{\hat{A}}_{h_{1}+h_{2}-2})_{m+n}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(W^{\hat{B}}_{h_{2}})_{n}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!-\tilde{q}^{h_{1},h_{2},-1}(m,n)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{h_{1}+h_{2}-1})_{m+n}$
$\displaystyle=\\!$
$\displaystyle-\frac{i}{4}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{h_{1}+h_{2}-1})_{m+n}\
,$ $\displaystyle\big{[}(W_{h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!q^{h_{1},h_{2}+1,0}(m,r-\frac{1}{2})\,(Q_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}$
$\displaystyle=\\!$
$\displaystyle\Big{[}m(h_{2}+1-1)-(r-\frac{1}{2})(h_{1}-1)\Big{]}\,(Q_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W_{h_{1}})_{m},(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!q^{h_{1},h_{2}+1,0}(m,r-\frac{1}{2})\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}$
$\displaystyle=\\!$
$\displaystyle\Big{[}m(h_{2}+1-1)-(r-\frac{1}{2})(h_{1}-1)\Big{]}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(Q_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!q^{h_{1},h_{2}+1,0}(m,r-\frac{1}{2})\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}$
$\displaystyle=\\!$
$\displaystyle\Big{[}m(h_{2}+1-1)-(r-\frac{1}{2})(h_{1}-1)\Big{]}\,(Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{h_{1}})_{m},(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!\hat{q}^{h_{1},h_{2}+\frac{1}{2},-1}(m,r)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}$
(3.29) $\displaystyle=\\!$
$\displaystyle-\frac{i}{4}\,f^{\hat{A}\hat{B}\hat{C}}\,(Q^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
.$

Due to the weight $(h_{2}+\frac{1}{2})$ for the fermionic current rather than
$(h_{2}+\frac{3}{2})$, the shift in $h_{2}$ in the right hand side appears.
Note that the structure constant $q^{h_{1},h_{2}+1,-1}(m,r-\frac{1}{2})$
vanishes. In other words, the lowest dummy variable $h$ in the first three
equations of (3.5) starts with $h=0$. Moreover, the structure constant
$\check{q}^{h_{1},h_{2}+\frac{1}{2},-1}(m,r)$ in the last equation of (3.5)
vanishes. This algebra (3.29) with a rescaling of the structure constant
$f^{\hat{A}\hat{B}\hat{C}}$ was found in [8] previously. The point here is
that the present description is more transparent because the last three
commutator relations of (3.29) in [8] are introduced abstractly but in this
paper we prove that they can be obtained from the above ${\cal N}=1$
supersymmetric $W_{\infty}^{K}$ algebra by taking the vanishing $\lambda$
limit. Therefore, at least, the celestial holography between the above two-
dimensional symmetry algebra and the OPEs [49] from the supersymmetric
Einstein-Yang-Mills theory holds at vanishing $\lambda$ limit.

###  3.4 The seven OPEs

In order to present the above commutator relations in terms of OPEs, we need
to introduce the following quantity [40]

$\displaystyle
M_{h}^{h_{1},h_{2}}(m,n)\equiv\sum_{k=0}^{h+1}\,(-1)^{k}\,\left(\begin{array}[]{c}h+1\\\
k\\\ \end{array}\right)(2h_{1}-h-2)_{k}[2h_{2}-2-k]_{h+1-k}\,m^{h+1-k}\,n^{k}\
.$ (3.32)

The degree of this polynomial is given by $(h+1)$. Then the above seven
commutator relations can be written in terms of the OPEs (See also [50])

$\displaystyle W_{h_{1}}(\bar{z})\,W_{h_{2}}(\bar{w})\\!$ $\displaystyle=\\!$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,f^{h_{1},h_{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{W_{h_{1}+h_{2}-2-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle W_{h_{1}}(\bar{z})\,W^{\hat{A}}_{h_{2}}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,f^{h_{1},h_{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{W^{\hat{A}}_{h_{1}+h_{2}-2-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle
W^{\hat{A}}_{h_{1}}(\bar{z})\,W^{\hat{B}}_{h_{2}}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,\tilde{f}^{h_{1},h_{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{-\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{h_{1}+h_{2}-2-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle+\\!$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,f^{h_{1},h_{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,$
$\displaystyle\times\\!$
$\displaystyle\Bigg{[}\frac{\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{h_{1}+h_{2}-2-h}(\bar{w})+\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,W_{h_{1}+h_{2}-2-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}\
+\cdots\ ,$ $\displaystyle
W_{h_{1}}(\bar{z})\,Q_{h_{2}+\frac{1}{2}}(\bar{w})\\!$ $\displaystyle=\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,(-1)^{h-1}\,f^{h_{1},h_{2}+1,h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{Q_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle
W_{h_{1}}(\bar{z})\,Q^{\hat{A}}_{h_{2}+\frac{1}{2}}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,(-1)^{h-1}\,f^{h_{1},h_{2}+1,h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle
W^{\hat{A}}_{h_{1}}(\bar{z})\,Q_{h_{2}+\frac{1}{2}}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-3}\,\lambda^{h}\,(-1)^{h-1}\,\Bigg{[}f^{h_{1},h_{2}+1,h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\frac{Q^{\hat{A}}_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
,$ $\displaystyle
W^{\hat{A}}_{h_{1}}(\bar{z})\,Q^{\hat{B}}_{h_{2}+\frac{1}{2}}(\bar{w})\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,\hat{f}^{h_{1},h_{2}+\frac{1}{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}$
(3.33) $\displaystyle+\\!$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h-1}\,\check{f}^{h_{1},h_{2}+\frac{1}{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,$
$\displaystyle\times\\!$
$\displaystyle\Bigg{[}\frac{\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})+\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,Q_{h_{1}+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\
\Bigg{]}$ $\displaystyle+\\!$
$\displaystyle\sum_{h=-1}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h}\,\dot{f}^{h_{1},h_{2}+\frac{1}{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,\Bigg{[}\frac{\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{h_{1}-1+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}$
$\displaystyle+\\!$
$\displaystyle\sum_{h=0}^{h_{1}+h_{2}-4}\,\lambda^{h}\,(-1)^{h}\,\bar{f}^{h_{1},h_{2}+\frac{1}{2},h}(\bar{\partial}_{\bar{z}},\bar{\partial}_{\bar{w}})\,$
$\displaystyle\times\\!$
$\displaystyle\Bigg{[}\frac{\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{h_{1}-1+h_{2}-\frac{3}{2}-h}(\bar{w})+\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,Q_{h_{1}-1+h_{2}-\frac{3}{2}-h}(\bar{w})}{(\bar{z}-\bar{w})}\Bigg{]}+\cdots\
.$

The various differential operators coming from the structure constants act on
the two complex coordinates $(\bar{z},\bar{w})$. The currents in the right
hand sides do depend on the coordinate $\bar{w}$.

From the structure constants where the quantity $N_{h}^{h_{1},h_{2}}(m,n)$ in
(2.9) is replaced by the quantity $M_{h}^{h_{1},h_{2}}(m,n)$ in (3.32)

$\displaystyle f_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{(-1)^{h}}{4(h+2)!}\Bigg{[}(h_{1}-1)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,0)$
(3.34) $\displaystyle-\\!$
$\displaystyle(h_{1}-h-3)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,\textstyle{-\frac{1}{2}})\Bigg{]}\,M^{h_{1},h_{2}}_{h}(m,r)\
,$ $\displaystyle f_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)\\!$
$\displaystyle\equiv\\!$
$\displaystyle\frac{-1}{4(h+2)!}\Bigg{[}(h_{1}-h-2)\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,0)-(h_{1})\,\phi^{h_{1},h_{2}+\frac{1}{2}}_{h+1}(0,\textstyle{-\frac{1}{2}})\Bigg{]}\,M^{h_{1},h_{2}}_{h}(m,r)\
,$

the previous structure constants together with (3.34) can be expressed as
follows:

$\displaystyle f^{h_{1},h_{2},h}(m,n)\\!$ $\displaystyle\equiv\\!$
$\displaystyle f_{\mathrm{B}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle+\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{B}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})$
$\displaystyle+\\!$ $\displaystyle
f_{\mathrm{F}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{F}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})\
,$ $\displaystyle\tilde{f}^{h_{1},h_{2},h}(m,n)$ $\displaystyle\equiv$
$\displaystyle f_{\mathrm{B}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle+\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{B}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})$
$\displaystyle-\\!$ $\displaystyle
f_{\mathrm{F}}^{h_{1},h_{2}-\frac{1}{2},h}(m,n+\frac{1}{2})$
$\displaystyle+\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{F}}^{h_{1}-1,h_{2}-\frac{1}{2},h-1}(m,n+\frac{1}{2})\
,$ $\displaystyle\hat{f}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle-
f_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)+f_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\ ,$
$\displaystyle\check{f}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle
f_{\mathrm{B}}^{h_{1},h_{2},h}(m,r)+f_{\mathrm{F}}^{h_{1},h_{2},h}(m,r)\ ,$
$\displaystyle\dot{f}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle-\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{B}}^{h_{1}-1,h_{2},h}(m,r)$
$\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{F}}^{h_{1}-1,h_{2},h}(m,r)\
,$ $\displaystyle\bar{f}^{h_{1},h_{2},h}(m,r)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{B}}^{h_{1}-1,h_{2},h}(m,r)$
(3.35) $\displaystyle-\\!$
$\displaystyle\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,f_{\mathrm{F}}^{h_{1}-1,h_{2},h}(m,r)\
.$

In Appendix $B$, we present the explicit OPEs between the currents $W_{4}$,
$W_{4}^{\hat{A}}$, $Q_{\frac{7}{2}}$ and $Q_{\frac{7}{2}}^{\hat{A}}$. On the
one hand, from the construction of free field realization, the OPEs can be
obtained from either by hands or by Thielemans package [52] for fixed weights
$h_{1}$ and $h_{2}$. On the other hand, we can write down (3.33) explicitly by
substituting the structure constants which are differential operators into
(3.33). The crucial point here is that from the transformation of (3.35) to
the ones in (3.33) where $m\rightarrow\bar{\partial}_{\bar{z}}$ and
$n,r\rightarrow\bar{\partial}_{\bar{w}}$, we should keep only the terms having
a degree $(h+1)$ of the polynomial. In general, the structure constants in
(3.35) do have the terms having a power sum of two variables less than the
above $(h+1)$ of the polynomial, contrary to the case of (3.34). We will
provide some details in Appendix $B$.

###  3.5 The possible realization in the ${\cal N}=1$ supersymmetric Einstein-
Yang-Mills theory

It is known, in [5], that the leading OPE for two positive helicity gluons is
given by the simple pole in the holomorphic sector with Euler beta function
whose arguments depend on the two conformal weights appearing on the left hand
side. They consider the scattering states in MHV (Maximally Helicity
Violating) tree amplitudes. The celestial amplitudes can be written as the
massless $n$ particle amplitudes, which depend on the energies and the points
on the celestial sphere, in the Mellin space. These amplitudes can be
interpreted as correlation functions of $n$ primary operators on the celestial
sphere. The simplest nontrivial scattering process occurs in the $n=4$ gluons.
Then the leading order behavior for the above massless four particle
amplitudes can be obtained by taking the holomorphic collinear limit for the
positive helicity (outgoing) gluons. Then it is possible to relate the full
$n=4$ amplitude to $n=3$ amplitude and the corresponding celestial four point
correlation function can be expressed as an integral over some integral
parameter where the integrand contains the three point correlation function.
Moreover the OPE of two positive helicity (outgoing) gluons can be described
as a conformal block including all the antiholomorphic descendants. Then the
leading OPE for two positive helicity gluons we mentioned before can be
further simplified after performing a Taylor expansion. Finally, the OPE of
the conformally soft gluon operators takes the simple pole in the holomorphic
sector with finite sum of antiholomorphic derivatives acting on the the second
gluon appearing on the left hand side of the OPE. In this soft limit, the
poles appearing in the OPE coefficients (Euler beta function) disappear and
the rescaled soft gluon operators with specific weights occur in the OPE.

For the supersymmetric Einstein-Yang-Mills theory which is a generalization of
previous paragraph, from the collinear limit of the respective Feynman matrix
element, the OPEs can be obtained by performing the Mellin transforms [49].
For example, the OPE of the conformal primary graviton and the conformal
primary gravitino of arbitrary weights takes the simple pole in the
holomorphic sector and the OPE coefficient is given by Euler beta function
which depends on the two previous weights on the left hand side of the OPE. By
taking the soft limit, this OPE between the soft positive helicity graviton
and the soft positive helicity gravitino can be written in terms of binomial
coefficient which depends on the two weights of the soft operators on the left
hand side and dummy variable for the finite sum for the antiholomorphic
derivatives acting on the soft positive helicity gravitino [8]. The structure
of this OPE looks like the one in previous case between two soft gluon
operators in the sense that the numerical values appearing in the binomial
coefficient and the power of the difference in the antiholomorphic coordinates
are little different.

The MHV gluon amplitudes in previous paragraph are the simplest amplitudes in
Yang-Mills theory. The next-to-simplest amplitudes, Next-to-MHV or NMHV sector
is studied in [20] based on the non minimal couplings of gluons and gravitons
by following the work of [32]. From the six-point NMHV analysis, the amplitude
is no longer a finite polynomial in the complex coordinates (and its complex
conjugated ones) of the soft particle. This leads to the fact that the lower
limits of the holomorphic and the antiholomorphic mode expansions of the soft
graviton, gravitino, gluon and gluino are given by $-\infty$ instead of finite
values in MHV sector. Furthermore, the upper limits in the dummy variable
appearing on the right hand sides of the OPEs between these soft operators are
not finite values but $\infty$. This allows us to obtain the mode dependent
function, which also depends on the three weights, when we express the OPEs in
terms of the commutators between the soft operators.

According to [32, 20], the OPE between the soft positive helicity graviton,
where the weights are $h_{1}=\frac{k-2}{2}$ and $h_{2}=\frac{l-2}{2}$, is
given by, after taking the soft limit,

$\displaystyle H^{k}(z_{1},\bar{z}_{1})\,H^{l}(z_{2},\bar{z}_{2})\\!$
$\displaystyle=\\!$
$\displaystyle-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}2-2h-k-
l-n\\\
1-h-l\end{array}\right)\,\frac{\bar{z}_{12}^{n+h+1}}{n!}\,\bar{\partial}^{n}\,H^{k+l+h}(z_{2},\bar{z}_{2})$
(3.38) $\displaystyle+\\!$ $\displaystyle\cdots\ .$ (3.39)

We can check the weights in the antiholomorphic sector both sides 999We denote
the antiholomorphic weights as $h_{1},h_{2}$ and $h$ without barred notation
by taking the notations in previous sections. For the notation of derivative,
we are using the barred notation as in $\bar{\partial}_{\bar{z}}$. Taking the
holomorphic and antiholomorphic expansions for the above soft graviton current
we obtain:
$H^{k}(z,\bar{z})=\sum_{n=-\infty}^{\frac{2-k}{2}}\,\frac{H_{n}^{k}(z)}{\bar{z}^{n+\frac{k-2}{2}}}=\sum_{m=-\infty}^{\frac{-2-k}{2}}\,\sum_{n=-\infty}^{\frac{2-k}{2}}\,\frac{H_{m,n}^{k}}{z^{m+\frac{k+2}{2}}\,\bar{z}^{n+\frac{k-2}{2}}}$.
The operator $H_{m,n}^{k}$ is therefore independent of $z$ and $\bar{z}$ and
we focus on the case where the mode $m$ is equal to $(1-h)$:
$\hat{H}_{n}^{k}\equiv H_{m=-\frac{k}{2},n}^{k}$.. The left hand side has the
weight $\frac{k-2}{2}+\frac{l-2}{2}$ while the right hand side has the weight
$-(n+h+1)+n+\frac{k+l-2}{2}+h$ from the last three factors. We observe that
for the $n$ and $h$ dependences in (3.39), there are no additional weight
contributions. In other words, they are cancelled each other and the inclusion
of $h$ does not change the weight in the form of (3.39). Note that there
exists the $h$-dependence on the right hand side of (3.39) and for $h=0$, we
reproduce the result of [5] with proper upper limit for the summation variable
$n$. It turns out that by performing the explicit contour integrals presented
in [20], the following commutator relation from (3.39) is satisfied 101010 In
[20], there are two important identities around their equations $(3.8)$ and
$(3.10)$. Their $p$ corresponds to our $(h+1)$. In their equation $(3.8)$, we
see that the summation over dummy variable with the upper bound $p$ reflects
the mode dependent function $N^{h_{1},h_{2}}_{h}(m,n)$ (2.9). They manage to
express the infinite sum over their $\alpha$ variable in terms of a product of
binomial coefficients. The final result is given by their equation (3.16)
after changing the nontrivial transformation in the mode expansion [6]. Note
that the general expression for the binomial coefficient is given by
$\left(\begin{array}[]{c}-2h_{1}-2h_{2}-2(1+h)-n\\\
-2h_{2}-(1+h)\end{array}\right)$ and their results will be used later.

$\displaystyle\big{[}(\hat{w}_{h_{1}})_{m},(\hat{w}_{h_{2}})_{n}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!(-1)^{h+1}\,N^{h_{1},h_{2}}_{h}(m,n)\,(\hat{w}_{h_{1}+h_{2}-2-h})_{m+n}\
.$ (3.40)

Here $\hat{w}_{h}$ is the rescaled weight-$h$ current and does not depend on
the complex coordinate [20, 8]. The mode dependent function
$N^{h_{1},h_{2}}_{h}(m,n)$ is given by the equation (2.9).

By using the two relations in (3.39) and (3.40) with associated other
relations for the gravitino, gluon and gluino, we would like to construct the
OPEs in the supersymmetric Einstein-Yang-Mills theory related to the previous
seven commutator relations. Note how the weights $(h_{1},h_{2})$ and the modes
$(m,n)$ of the left hand side appear on the right hand side of (3.40).

####  3.5.1 The OPE between the soft positive helicity graviton and the soft
positive helicity gravitino

At vanishing deformation parameter, the OPE of the conformal primary graviton
and the conformal primary gravitino of arbitrary weights is given by [49]
111111 We have ${\cal O}_{\Delta_{1},+2}(z_{1},\bar{z}_{1})\,{\cal
O}_{\Delta_{2},+\frac{3}{2}}(z_{2},\bar{z}_{2})=-\frac{\frac{\kappa}{2}}{z_{12}}\,\sum_{n=0}^{\infty}\,B(\Delta_{1}-1+n,\Delta_{2}-\frac{1}{2})\,\frac{\bar{z}_{12}^{n+1}}{n!}\,\bar{\partial}^{n}\,{\cal
O}_{\Delta_{1}+\Delta_{2},+\frac{3}{2}}(z_{2},\bar{z}_{2})+\cdots$. . Then the
question is how the contributions from the nonzero deformation parameter occur
on the right hand side of this OPE.

  * i) At least, we should have the OPE structure found by [32] which is the fact that the OPE contains the $h$ dependence in the $\bar{z}_{12}$, the binomial coefficient and the weight of the operator appearing on the right hand side. These OPE coefficients are also obtained from the analysis in the collinear limits. See also the equation $(3.2)$ of [32] in which the addition of fermions is also valid.

ii) As mentioned before (in the non minimal couplings of gluons and
gravitons), after taking the soft limit for the OPE in i) in order to absorb
the infinite number of poles appearing in the binomial coefficient, we also
require that there is no restriction on the lower limits of the holomorphic
and antiholomorphic mode expansions of the soft operators [20]. See also the
equation $(A.17)$ of [20].

iii) We also should sum over all the contributions from each fixed $h$ on the
right hand side of the OPE. Moreover, due to the construction of ${\cal N}=1$
supersymmetric theory in previous section, we should also consider the
contributions from the antiholomorphic derivatives acting on the soft currents
on the left hand sides of the OPEs. In some sense, we make the supersymmetric
generalization of the work of [20].

Let us consider the case where one of the soft current on the left hand side
in the OPE contains the fermionic current.

First of all, the OPE between the conformally soft positive helicity gravitons
and the conformally soft positive helicity gravitinos, where the weights in
the antiholomorphic sector are given by $h_{1}=\frac{k-2}{2}$ and
$h_{2}=\frac{l-\frac{3}{2}}{2}$, can be generalized to the following
expression

$\displaystyle
H^{k}(z_{1},\bar{z}_{1})\,I^{l}(z_{2},\bar{z}_{2})=-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{h=0}^{h_{1}+h_{2}-3}\,(-1)^{h+1}\,\lambda^{h}\,\Bigg{[}q_{B}^{h_{1},h_{2}+\frac{1}{2},h}+q_{F}^{h_{1},h_{2}+\frac{1}{2},h}\Bigg{]}\,$
$\displaystyle\times\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}\frac{3}{2}-2h-k-
l-n\\\
\frac{1}{2}-h-l\end{array}\right)\,\frac{\bar{z}_{12}^{n+h+1}}{n!}\,\bar{\partial}^{n}\,I^{k+l+h}(z_{2},\bar{z}_{2})\
$ (3.43)
$\displaystyle-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{h=0}^{h_{1}+h_{2}-3}\,(-1)^{h}\,\lambda^{h}\,\Bigg{[}\frac{2(h_{1}-2)}{2(h_{1}-2)+1}\,q_{B}^{h_{1}-1,h_{2}+\frac{1}{2},h-1}-\frac{2(h_{1}-2)+2}{2(h_{1}-2)+1}\,q_{F}^{h_{1}-1,h_{2}+\frac{1}{2},h-1}\Bigg{]}\,$
$\displaystyle\times\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}\frac{3}{2}-2(h-1)-(k-2)-l-n\\\
\frac{1}{2}-(h-1)-l\end{array}\right)\,\frac{\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}}{n!}\,\bar{\partial}^{n}\,I^{(k-2)+l+(h-1)}(z_{2},\bar{z}_{2})$
(3.46) $\displaystyle+\cdots\ .$ (3.47)

The numerical factor $\frac{3}{2}$ in the first line of the first binomial
coefficient is given by $(2+\frac{3}{2})$ (which is the sum of numerical
numbers with minus sign in the numerator of previous weights) minus $2$. The
numerical factor $\frac{1}{2}$ in the second line of the first binomial
coefficient is given by $\frac{3}{2}$ (which is the numerical number for the
gravitino with minus sign in the numerator of previous weights) minus $1$
121212 Or we can use the formula in the footnote 10.. The first binomial
coefficient above can be obtained from (3.39) by taking the shift
$l\rightarrow(l+\frac{1}{2})$.

In the above, we multiplied the factor $(-1)^{h+1}$ in order to absorb
$(-1)^{h+1}$ factor on the right hand side of (3.40). Here the graviton
current is replaced by the gravitino current on the left hand side of the OPE
in (3.39) 131313That is,
$H^{k}(z_{1},\bar{z}_{1})\,I^{l}(z_{2},\bar{z}_{2})=-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}\frac{3}{2}-2h-k-
l-n\\\
\frac{1}{2}-h-l\end{array}\right)\,\frac{\bar{z}_{12}^{n+h+1}}{n!}\,\bar{\partial}^{n}\,I^{k+l+h}(z_{2},\bar{z}_{2})+\cdots\
.$ For $h=0$ with proper upper limit of dummy variable $n$, we observe the
result of [8].. Now we sum over the dummy variable $h$ from $0$ to
$(h_{1}+h_{2}-3)$ together with the structure constant, which depends on
$h_{1}$, $h_{2}$ and $h$ only without mode-dependent factor in (2.3), in order
to obtain the first two lines of (3.47). In other words, from the first
equation of (3.5) and (3.4), the full structure constant contains
$q_{B}^{h_{1},h_{2}+\frac{1}{2},h}(m,n)+q_{F}^{h_{1},h_{2}+\frac{1}{2},h}(m,n)$,
which is redefined as
$(q_{B}^{h_{1},h_{2}+\frac{1}{2},h}+q_{F}^{h_{1},h_{2}+\frac{1}{2},h})\,N_{h}^{h_{1},h_{2}+\frac{1}{2}}(m,n)$
together with (2.3). Then we can apply the property of (3.40) to the
commutator, the first equation of (3.5), and arrive at the first two lines of
above OPE. Here the mode-independent factor,
$(q_{B}^{h_{1},h_{2}+\frac{1}{2},h}+q_{F}^{h_{1},h_{2}+\frac{1}{2},h})$, can
be combined to other $h$-dependent factor, $(-1)^{h+1}\,\lambda^{h}$, inside
the $h$ summation. We observe that the weights in the antiholomorphic sector
are preserved at both sides because the powers of the dummy variable $n$ and
the weight $h$ are the same as before.

In the third and fourth lines of (3.47), we use the property between the mode
of current and the current itself in (3.1) and (3.17). Note that there exists
minus sign between them. Recall the remaining full structure constant contains
$\frac{2(h_{1}-2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{B}^{h_{1}-1,h_{2}+\frac{1}{2},h-1}(m,n)-\frac{(2(h_{1}-2)+2)(m+(h_{1}-2)+1)}{2(h_{1}-2)+1}\,q_{F}^{h_{1}-1,h_{2}+\frac{1}{2},h-1}(m,n)$.
As we did before, we factorize the mode-dependent term
$(m+(h_{1}-2)+1)\,N^{h_{1}-1,h_{2}+\frac{1}{2}}_{h-1}(m,n)$ and the other
mode-independent terms. The partial derivative $\bar{\partial}_{\bar{z}_{1}}$
of $H^{k-2}(z_{1},\bar{z}_{1})$ on the left hand side appears in the form of
$\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}$ on the right hand
side of the OPE. Further shifts in the weights $h_{1}$ and $h$ in the
structure constant affect the shifts in the corresponding the $k$ and the
weight $h$ in the OPE. Note that the weight in the antiholomorphic sector in
the last line of (3.47) behaves correctly because two additional weight $2$ in
the derivative term $\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}$
is cancelled by the corresponding additional $-2$ coming from the factors
$(k-2)$ and $(h-1)$, compared to the second line of (3.47).

Then by performing the contour integrals as in [20] (See also [5, 8]), the
above OPE (3.47), from the analysis of two previous paragraphs, provides the
first equation of (3.5) precisely 141414 In an expression of
$\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}$, we can write down
this as $(n+(h-1)+1)\,\bar{z}_{12}^{n+(h-1)}$ after a differentiation.
Compared to the one without a derivative, there is an extra factor
$(n+(h-1)+1)$ with different power of $\bar{z}_{12}$. Now we can combine this
with the binomial coefficient $\left(\begin{array}[]{c}n+(h-1)\\\
-m-h_{1}\end{array}\right)$ appearing in the $\bar{z}_{1}$ integral when we
calculate the commutator from the OPE. Then it is easy to check that the
following relation is satisfied $(n+(h-1)+1)\left(\begin{array}[]{c}n+(h-1)\\\
-m-h_{1}\end{array}\right)=-(m+h_{1}-1)\,\Bigg{[}\frac{(n+(h-1)+1)!}{(n+(h-1)+m+h_{1})!(1-m-h_{1})!}\Bigg{]}=-(m+h_{1}-1)\,\Bigg{[}\frac{(n+(h-1)+1)!}{(n+(h-1)+1+m+h_{1})!(-m-h_{1})!}\Bigg{]}\Bigg{|}_{h_{1}\rightarrow
h_{1}-1}$. This implies that the remaining calculation is the same as the one
in [20] with an overall factor $-(m+h_{1}-1)=-(m+(h_{1}-2)+1)$ which appears
in the second term of the first structure constant in (3.4) as we expected. Of
course, the factor $(-\bar{z}_{2})^{n+(h-1)+m+h_{1}}$ coming from the
$\bar{z}_{1}$ integral can be written as
$(-\bar{z}_{2})^{n+(h-1)+1+m+(h_{1}-1)}=(-\bar{z}_{2})^{n+(h-1)+1+m+h_{1}}\Bigg{|}_{h_{1}\rightarrow
h_{1}-1}$. Moreover, the binomial coefficient is
$\left(\begin{array}[]{c}\frac{3}{2}-2(h-1)-(k-2)-l-n\\\
\frac{1}{2}-(h-1)-l\end{array}\right)=\left(\begin{array}[]{c}\frac{3}{2}-2(h-1)-k-l-n\\\
\frac{1}{2}-(h-1)-l\end{array}\right)\Bigg{|}_{h_{1}\rightarrow h_{1}-1}$. Of
course, $I^{(k-2)+l+(h-1)}=I^{k+l+(h-1)}\Bigg{|}_{h_{1}\rightarrow h_{1}-1}$.
. Therefore, we have determined the OPE between the soft positive helicity
graviton and the soft positive helicity gravitino which can be obtained from
(or leads to) the corresponding commutator in two dimensions studied in
previous section. We present the details for other OPEs including the soft
positive helicity gravitinos or soft positive helicity gluinos in Appendix $C$
explicitly where other structure constants (3.6) can be used appropriately.

####  3.5.2 The OPE between the soft positive helicity gravitons

Let us consider the second example, which is more nontrivial, for the
appearance of the two dimensional symmetry algebra in the four dimensional
Einstein-Yang-Mills theory. Based on the three features i), ii) and iii), in
previous subsection, we can write down the corresponding OPE for soft positive
helicity graviton in the supersymmetric Einstein-Yang-Mills theory, by
comparing (3.40) with the first equation of (3.2), as follows:

$\displaystyle
H^{k}(z_{1},\bar{z}_{1})\,H^{l}(z_{2},\bar{z}_{2})=-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{h=1,\mbox{\footnotesize
odd}}^{h_{1}+h_{2}-4}\,(-1)^{h}\,\lambda^{h}\,\Bigg{[}f_{1}^{h_{1}-1,h_{2},h-1}$
$\displaystyle\times\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}2-2(h-1)-(k-2)-l-n\\\
1-(h-1)-l\end{array}\right)\,\frac{\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}}{n!}\,\bar{\partial}^{n}\,H^{(k-2)+l+(h-1)}(z_{2},\bar{z}_{2})$
(3.50)
$\displaystyle+f_{2}^{h_{1},h_{2}-1,h-1}\,\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}2-2(h-1)-k-(l-2)-n\\\
1-(h-1)-(l-2)\end{array}\right)\,$ (3.53)
$\displaystyle\times\frac{1}{n!}\,\bar{\partial}_{\bar{z_{2}}}\,\big{[}\bar{z}_{12}^{n+(h-1)+1}\,\bar{\partial}^{n}\,H^{k+(l-2)+(h-1)}(z_{2},\bar{z}_{2})\big{]}\Bigg{]}-\frac{\kappa}{2}\,\frac{1}{z_{12}}\,\sum_{h=0,\mbox{\footnotesize
even}}^{h_{1}+h_{2}-4}\,(-1)^{h+1}\,\lambda^{h}\,\Bigg{[}f_{3}^{h_{1},h_{2},h}$
$\displaystyle\times\,\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}2-2h-k-
l-n\\\
1-h-l\end{array}\right)\,\frac{\bar{z}_{12}^{n+h+1}}{n!}\,\bar{\partial}^{n}\,H^{k+l+h}(z_{2},\bar{z}_{2})$
(3.56)
$\displaystyle+f_{4}^{h_{1}-1,h_{2}-1,h-2}\,\sum_{n=0}^{\infty}\,\left(\begin{array}[]{c}2-2(h-2)-(k-2)-(l-2)-n\\\
1-(h-2)-(l-2)\end{array}\right)\,$ (3.59)
$\displaystyle\times\frac{1}{n!}\,\bar{\partial}_{\bar{z_{1}}}\,\bar{\partial}_{\bar{z_{2}}}\,\big{[}\bar{z}_{12}^{n+(h-2)+1}\,\bar{\partial}^{n}\,H^{(k-2)+(l-2)+(h-2)}(z_{2},\bar{z}_{2})\big{]}\Bigg{]}+\cdots\
.$ (3.60)

Here we introduce the following quantities

$\displaystyle f_{1}^{h_{1}-1,h_{2},h-1}$ $\displaystyle\equiv$
$\displaystyle\Bigg{[}\frac{2(h_{1}-2)}{2(h_{1}-2)+1}\,\frac{(h_{1}+h_{2}-2-h)}{2(h_{1}+h_{2}-2-h)+1}\,p_{B}^{h_{1}-1,h_{2},h-1}$
$\displaystyle-\frac{(h_{1}+h_{2}-2-h-1)}{2(h_{1}+h_{2}-2-h-1)+1}\,\frac{2(h_{1}-2)+2}{2(h_{1}-2)+1}\,p_{F}^{h_{1}-1,h_{2},h-1}\Bigg{]},$
$\displaystyle f_{2}^{h_{1},h_{2}-1,h-1}$ $\displaystyle\equiv$
$\displaystyle\Bigg{[}\frac{2(h_{2}-2)}{2(h_{2}-2)+1}\,\frac{(h_{1}+h_{2}-2-h)}{2(h_{1}+h_{2}-2-h-1)+1}\,p_{B}^{h_{1},h_{2}-1,h-1}$
$\displaystyle-\frac{(h_{1}+h_{2}-2-h-1)}{2(h_{1}+h_{2}-2-h-1)+1}\,\frac{2(h_{2}-2)+2}{2(h_{2}-2)+1}\,p_{F}^{h_{1},h_{2}-1,h-1}\Bigg{]},$
$\displaystyle f_{3}^{h_{1},h_{2},h}$ $\displaystyle\equiv$
$\displaystyle\Bigg{[}\frac{(h_{1}+h_{2}-2-h)}{2(h_{1}+h_{2}-2-h-1)+1}p_{B}^{h_{1},h_{2},h}+\frac{(h_{1}+h_{2}-2-h-1)}{2(h_{1}+h_{2}-2-h-1)+1}p_{F}^{h_{1},h_{2},h}\Bigg{]},$
$\displaystyle f_{4}^{h_{1}-1,h_{2}-1,h-2}$ $\displaystyle\equiv$
$\displaystyle\Bigg{[}\frac{2(h_{1}-2)}{2(h_{1}-2)+1}\,\frac{2(h_{2}-2)}{2(h_{2}-2)+1}\,\frac{(h_{1}+h_{2}-2-h)}{2(h_{1}+h_{2}-2-h-1)+1}\,p_{B}^{h_{1}-1,h_{2}-1,h-2}$
$\displaystyle+\frac{2(h_{1}-2)+2}{2(h_{1}-2)+1}\,\frac{2(h_{2}-2)+2}{2(h_{2}-2)+1}\,\frac{(h_{1}+h_{2}-2-h-1)}{2(h_{1}+h_{2}-2-h-1)+1}\,p_{F}^{h_{1}-1,h_{2}-1,h-2}\Bigg{]}.$

Let us describe how we obtain this result. Now we return to the OPE (3.60) for
the soft positive helicity gravitons. We multiply the factor $(-1)^{h+1}$ into
the relation (3.39) and sum over the variable $h$ from $0$ to
$(h_{1}+h_{2}-4)$ together with the structure constant which depends on
$h_{1}$, $h_{2}$ and $h$ without mode-dependent factor in (2.3) as before. For
the mode-dependent part, we want to use the previous relation (3.40). In the
first three lines of (3.60), we again use the property between the mode of
current and the current itself in (3.1) and (3.17). The partial derivative
$\bar{\partial}_{\bar{z}_{1}}$ of $H^{k-2}(z_{1},\bar{z}_{1})$ appears in the
form of $\bar{\partial}_{\bar{z_{1}}}\,\bar{z}_{12}^{n+(h-1)+1}$ on the right
hand side of the OPE. The mode-independent coefficients come from the second
and the fourth terms of the first equation in (C.2) with (3.5.2). Further
shifts in the weights $h_{1}$ and $h$ in the structure constant affect the
shifts in the corresponding $k$ and $h$. Note that the weight in the
antiholomorphic sector here behaves correctly. See also the footnote 14.

In the next line, the $\bar{\partial}_{\bar{z}_{2}}$ acts on both the
coordinate $\bar{z}_{12}$ and $H^{k+(l-2)+(h-1)}(z_{2},\bar{z}_{2})$
(corresponding to $\bar{\partial}_{\bar{z}_{2}}$ of
$H^{l-2}(z_{2},\bar{z}_{2})$ in the left hand side) 151515From the
contribution of
$\bar{\partial}_{\bar{z}_{2}}\,\bar{z}_{12}^{n^{\prime}+(h-1)+1}$, there is an
extra factor $-(n^{\prime}+(h-1)+1)$ with $\bar{z}_{12}^{n^{\prime}+(h-1)}$
where $n^{\prime}$ is a previous dummy variable $n$. By recalling the
$\bar{z}_{1}$ contour integral, we have
$-(n^{\prime}+(h-1)+1)\,\left(\begin{array}[]{c}n^{\prime}+(h-1)\\\
-m-h_{1}\end{array}\right)\,(-1)^{n^{\prime}+(h-1)+m+h_{1}}$ where
$h_{1}=\frac{k-2}{2}$. Then this can be rewritten as
$(n^{\prime}+(h-1)+1+m+h_{1})\,\left(\begin{array}[]{c}n^{\prime}+(h-1)+1\\\
-m-h_{1}\end{array}\right)\,(-1)^{n^{\prime}+(h-1)+1+m+h_{1}}$. Note that the
power of $\bar{z}_{12}$ is $n^{\prime}+(h-1)$. This implies that there exists
a factor $(n^{\prime}+(h-1)+1+m+h_{1})$ in the presence of
$\bar{\partial}_{\bar{z}_{2}}$. Of course, the factor
$(\bar{z}_{2})^{n^{\prime}+(h-1)+m+h_{1}+n+h_{2}-1}$ coming from the
$\bar{z}_{1}$ integral and the exponent of $\bar{z}_{2}$ in the contour
integral can be written as
$(\bar{z}_{2})^{n^{\prime}+(h-1)+1+m+h_{1}+n+(h_{2}-1)-1}$ which can be
obtained by taking $h_{2}\rightarrow(h_{2}-1)$ from the expression without a
derivative. Furthermore, from the contribution of
$\bar{\partial}_{\bar{z}_{2}}\,H^{k+(l-2)+(h-1)}(z_{2},\bar{z}_{2})$, we have
the extra factor $(-m-n-h_{1}-(h_{2}-1)-(h-1)-1-n^{\prime})$ (with
$h_{2}=\frac{l-2}{2}$) which can be obtained by taking
$h_{2}\rightarrow(h_{2}-1)$ from the expression without a derivative factor.
The exponent of $\bar{z}_{2}$ is given by
$(\bar{z}_{2})^{n^{\prime}+(h-1)+1+m+h_{1}+n+h_{2}-1}$ originally and from the
above derivative $\bar{\partial}_{\bar{z}_{2}}$ we have an additional
$\bar{z}_{2}^{-1}$, compared to the case without this derivative. Note that
the power of $\bar{z}_{12}$ is $n^{\prime}+(h-1)+1$. Therefore, the above
power can be written as $n^{\prime}+(h-1)+1+m+h_{1}+n+(h_{2}-1)-1$ which is
obtained by taking $h_{2}\rightarrow h_{2}-1$. The binomial coefficient can be
shifted similarly as in the footnote 14. Then finally we are left with the
factor $[n^{\prime}+(h-1)+1+m+h_{1}]+[-m-n-
h_{1}-(h_{2}-1)-(h-1)-1-n^{\prime}]=-n-(h_{2}-1)$ by adding these two
contributions. . In this case, the mode-independent coefficients come from the
first and the third terms of the first equation in (C.2).

For the remaining lines, we describe similarly by applying the first and the
third terms in the second equation of (C.2) without any derivatives in the
summation over even weight $h$. Finally, the second and the fourth terms in
the second equation of (C.2) with two derivatives play the role in the second
summation over even weight $h$ 161616We can analyze the action of
$\bar{\partial}_{\bar{z}_{1}}\,\bar{\partial}_{\bar{z}_{2}}$ appearing at the
end of (3.60) by taking the procedures in the footnote 14 and the footnote
15.. The additional weight-$2$ from the two derivatives is cancelled by those
from the power of $(k-2)$ and $(l-2)$ in the soft current.

Then by performing the contour integrals as in [20] similarly (See also [5,
8]), the above OPEs (3.47) and (3.60) provide the first equations of (3.5) and
(3.2) respectively. We can extract other five OPEs similarly by taking into
account of the weights for the soft graviton, gravitino, gluon and gluino in
the binomial coefficient above. We present the details in Appendix $C$
explicitly. In particular, the useful structure constants
$\tilde{q}^{h_{1},h_{2},h}(m,n)$ we are using, which appear in the third
equation of (3.2), are given at the end of Appendix $C$. Therefore, we have
found the precise correspondence between the OPEs between the conformally soft
operators in the ${\cal N}=1$ supersymmetric Einstein-Yang-Mills theory and
the two dimensional symmetry algebra.

##  4  Conclusions and outlook

The main result of this paper can be summarized by (3.2) and (3.5) together
with (3.4) and (3.6). That is, we have determined the three commutator
relations by combining the the corresponding commutator relations from the
bosonic currents made of the bosonic free fields with the corresponding
commutator relations from the bosonic currents made of the fermionic free
fields. Similarly, the remaining four commutator relations can be obtained
from two different kinds of commutator relations. Their OPE version can be
found in (3.33) with (3.34) and (3.35). We observe that there are also the
nontrivial singular terms whose poles are greater than two and they play the
role of the contributions from the deformation.

By using the celestial holography, we have obtained the results, summarized by
(3.47), (3.60), Appendices (C.9), (C.16), (C.34), (C.47), and (C.84) in the
supersymmetric Einstein-Yang-Mills theory at nonzero deformation parameter.
The common behavior is as follows. There exist a simple pole in the
holomorphic sector, the nontrivial structure constants which depend on the
three weights, binomial coefficients containing the dummy variable also as
well as the three weights and the descendant fields associated with the second
soft currents on the left hand sides. Furthermore, after calculating the
various commutator relations by using these seven OPEs between the soft
currents, we have checked the above seven commutators in two dimensions
discussed in previous paragraph 171717So far, we have focused on the soft
currents as mentioned in the footnote 1. According to the findings of [32,
31], the leading tree level celestial OPEs from the cubic vertices of three
spinning massless particles contains the OPE coefficients given by Euler beta
function whose arguments are $(\Delta_{1}+s_{2}-s_{3}-1)$ and
$(\Delta_{2}+s_{1}-s_{3}-1)$ where $s_{i}$ is a helicity or spin. In
particular, in [31], it is found that all the previous known seven nontrivial
celestial OPEs are obtained from this general formula. We can ask whether the
OPEs between the hard operators, which do not satisfy the conditions for the
conformal dimensions in the footnote 1, provide any two dimensional symmetry
algebra similar to the ones in this paper or not. It seems that the infinite
number of poles appearing in the above Euler beta function can appear only for
the corresponding conformal dimensions of the soft operators. In other words,
for the conformal dimensions of the hard operators, we cannot cover the
infinite number of poles and therefore we cannot remove all of them..

We have not discussed about the implications of the ${\cal N}=2$
supersymmetric $W_{\infty}^{K,K}$ algebra in the section $2$. It would be
interesting to find out whether the possibility of the ${\cal N}=2$
supersymmetric $w_{\infty}^{K,K}$ algebra occurs or not by further
examination. In the context of $AdS_{3}/CFT_{2}$ correspondence, the previous
algebra in (2.1) is related to the case of vanishing ’t Hooft-like coupling
constant. Therefore, it is an open question how the another deformed case with
nonvanishing ’t Hooft-like coupling constant [56] will arise in the context of
the present paper. We expect that the currents from the free field realization
will depend on this nonzero coupling constant explicitly and they will become
the currents we have described in this paper by taking this coupling constant
to be zero.

Acknowledgments

We would like to thank M.H. Kim for discussions. This work was supported by
the National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT)(No. 2020R1F1A1066893).

##  Appendix A  The remaining (anti)commutator relations in the ${\cal N}=2$
supersymmetric $W_{\infty}^{K,K}$ algebra with $U(K)\times U(K)$ symmetry

In this Appendix, we present some details which are related to the contents in
section $2$.

###  A.1 The commutators between the bosonic and the other fermionic currents

We can multiply the generators into the fourth equation of (2.1) and obtain
the following commutator relations

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

It is easy to check the following reduced commutator relations by taking
(2.74)

$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(\bar{Q}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(\bar{Q}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\ ,$
$\displaystyle\big{[}(W_{\mathrm{F},h_{1}})_{m},(\bar{Q}^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(\bar{Q}^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(\bar{Q}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}\,(\bar{Q}^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{F},h_{1}})_{m},(\bar{Q}^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{i}{8}\,f^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}+\frac{1}{4}\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}$
(A.2) $\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(\bar{Q}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\Bigg{]}\
.$

###  A.2 The commutators between the other bosonic and the other fermionic
currents

The sixth equation of (2.1) with the addition of generators leads to the
following commutator relations

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

By taking (2.74), the following reduced commutator relations hold

$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(\bar{Q}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(\bar{Q}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\ ,$
$\displaystyle\big{[}(W_{\mathrm{B},h_{1}})_{m},(Q^{\hat{A}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(\bar{Q}^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(\bar{Q}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,(\bar{Q}^{\hat{A}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\
,$
$\displaystyle\big{[}(W^{\hat{A}}_{\mathrm{B},h_{1}})_{m},(Q^{\hat{B}}_{h_{2}+\frac{1}{2}})_{r}\big{]}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{i}{8}\,f^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}-\frac{1}{4}\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(\bar{Q}^{\hat{C}}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}$
(A.4) $\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(\bar{Q}_{h_{1}+h_{2}-\frac{1}{2}})_{m+r}\Bigg{]}\
.$

###  A.3 The anticommutators between the fermionic currents

Finally, the seventh equation of (2.1) provides the following anticommutator
relations

$\displaystyle\\{(Q_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-1}_{h=0}\,\lambda^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W_{\mathrm{F},h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-2}_{h=0}\,\lambda^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W_{\mathrm{B},h_{1}+h_{2}-h})_{r+s}\
,$ $\displaystyle+\\!$ $\displaystyle
K\,c_{Q\bar{Q}_{h_{1}+\frac{1}{2}}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}+\frac{1}{2}-1)}\delta_{r+s}\
,$
$\displaystyle\\{(Q_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}^{\hat{A}}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-1}_{h=0}\,\lambda^{h}\,\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W^{\hat{A}}_{\mathrm{F},h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-2}_{h=0}\,\lambda^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W^{\hat{A}}_{\mathrm{B},h_{1}+h_{2}-h})_{r+s}\
,$
$\displaystyle\\{(Q^{\hat{A}}_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-1}_{h=0}\,\lambda^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W^{\hat{A}}_{\mathrm{F},h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\,\sum^{h_{1}+h_{2}-2}_{h=0}\,\lambda^{h}o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,(W^{\hat{A}}_{\mathrm{B},h_{1}+h_{2}-h})_{r+s}\
,$
$\displaystyle\\{(Q^{\hat{A}}_{h_{1}+\frac{1}{2}})_{r},(\bar{Q}^{\hat{B}}_{h_{2}+\frac{1}{2}})_{s}\\}\\!$
$\displaystyle=$
$\displaystyle\\!\sum^{h_{1}+h_{2}-1}_{h=0}\,\lambda^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{F,h_{1}+h_{2}-h})_{r+s}$
(A.5) $\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-1}_{h=0}\,\lambda^{h}\,o_{\mathrm{F}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{F,h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(W_{F,h_{1}+h_{2}-h})_{r+s}\Bigg{]}$
$\displaystyle-\\!$
$\displaystyle\sum^{h_{1}+h_{2}-2}_{h=0}\,\lambda^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\frac{i}{2}\,f^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{B,h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\sum^{h_{1}+h_{2}-2}_{h=0}\,\lambda^{h}\,o_{\mathrm{B}}^{h_{1}+\frac{1}{2},h_{2}+\frac{1}{2},h}(r,s)\,\Bigg{[}\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,(W^{\hat{C}}_{B,h_{1}+h_{2}-h})_{r+s}$
$\displaystyle+\\!$
$\displaystyle\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,(W_{B,h_{1}+h_{2}-h})_{r+s}\Bigg{]}$
$\displaystyle+\\!$ $\displaystyle
c_{Q\bar{Q}_{h_{1}+\frac{1}{2}}}\,\delta^{\hat{A}\hat{B}}\,\delta^{h_{1}h_{2}}\,\lambda^{2(h_{1}+\frac{1}{2}-1)}\delta_{r+s}\
.$

Then the complete algebra consists of (2.34), (2.35), (2.36), (2.37), Appendix
(A.1), Appendix (A.3) and Appendix (A.5).

Note that there are no nontrivial reduced anticommutator relations after
taking (2.74). Then we have the complete results, (2.75), Appendix (A.2) and
Appendix (A.4).

##  Appendix B  The operator product expansions in the ${\cal N}=1$
supersymmetric $W_{\infty}^{K}$ algebra with $U(K)$ symmetry

In this Appendix, we present some details which are related to the contents in
section $3$.

###  B.1  The seven OPEs for fixed $h_{1}$ and $h_{2}$

It is straightforward to calculate the following OPEs by using the Thielemans
package [52] inside the mathematica [57]. We use the equations (2.13), (3.17)
and (3.27). For fixed $h_{1}$ and $h_{2}$, we perform each pole starting from
the highest order pole. For each pole, we can consider the possible terms
(descendant terms and new higher spin current). The higher spin currents are,
in general, not quasiprimary. Although the current $W_{h}(\bar{w})$ for fixed
$h$ does not appear in the particular pole, its derivative term
$\bar{\partial}\,W_{h}(\bar{w})$ appears at the next order pole. This
derivative term plays the role of the quasiprimary current of the
weight-$(h+1)$. Once we rearrange each OPE in terms of quasiprimary currents
then the standard expressions in the right hand sides appear 181818Note that
$W_{4},W_{4}^{\hat{A}},Q_{\frac{7}{2}}$ and $Q^{\hat{A}}_{\frac{7}{2}}$ are
not quasiprimary under the $W_{2}$ stress energy tensor which has zero central
charge. We can make them to be quasiprimary by adding the derivatives of
currents having lower weights. The weights for $Q_{\frac{7}{2}}$ and
$Q^{\hat{A}}_{\frac{7}{2}}$ are $4$ not $\frac{7}{2}$ after topological
twisting.. We have the following OPEs for fixed $h_{1}$ and $h_{2}$

$\displaystyle
W_{4}(\bar{z})\,W_{4}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,W_{2}(\bar{w})+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\frac{4608\lambda^{4}}{5}\,\bar{\partial}\,W_{2}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\bar{\partial}^{2}\,W_{2}+\frac{2304\lambda^{3}}{125}\,\bar{\partial}\,W_{3}+\frac{8256\lambda^{2}}{25}\,W_{4}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\bar{\partial}^{3}\,W_{2}+\frac{1152\lambda^{3}}{125}\,\bar{\partial}^{2}\,W_{3}+\frac{4128\lambda^{2}}{25}\,\bar{\partial}\,W_{4}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\bar{\partial}^{4}\,W_{2}+\frac{2304\lambda^{3}}{875}\,\bar{\partial}^{3}\,W_{3}+\frac{8256\lambda^{2}}{175}\,\bar{\partial}^{2}\,W_{4}+\frac{8\lambda}{15}\,\bar{\partial}\,W_{5}+6\,W_{6}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\bar{\partial}^{5}\,W_{2}+\frac{96\lambda^{3}}{175}\,\bar{\partial}^{4}\,W_{3}+\frac{344\lambda^{2}}{35}\,\bar{\partial}^{3}\,W_{4}+\frac{4\lambda}{15}\,\bar{\partial}^{2}\,W_{5}+3\,\bar{\partial}\,W_{6}\Bigg{]}(\bar{w})$
$\displaystyle+\cdots\ ,$ $\displaystyle
W_{4}(\bar{z})\,W^{\hat{A}}_{4}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,W^{\hat{A}}_{2}(\bar{w})+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\frac{4608\lambda^{4}}{5}\,\bar{\partial}\,W^{\hat{A}}_{2}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\bar{\partial}^{2}\,W^{\hat{A}}_{2}+\frac{2304\lambda^{3}}{125}\,\bar{\partial}\,W^{\hat{A}}_{3}+\frac{8256\lambda^{2}}{25}\,W^{\hat{A}}_{4}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\bar{\partial}^{3}\,W^{\hat{A}}_{2}+\frac{1152\lambda^{3}}{125}\,\bar{\partial}^{2}\,W^{\hat{A}}_{3}+\frac{4128\lambda^{2}}{25}\,\bar{\partial}\,W^{\hat{A}}_{4}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\bar{\partial}^{4}\,W^{\hat{A}}_{2}+\frac{2304\lambda^{3}}{875}\,\bar{\partial}^{3}\,W^{\hat{A}}_{3}+\frac{8256\lambda^{2}}{175}\,\bar{\partial}^{2}\,W^{\hat{A}}_{4}+\frac{8\lambda}{15}\,\bar{\partial}\,W^{\hat{A}}_{5}+6\,W^{\hat{A}}_{6}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\bar{\partial}^{5}\,W^{\hat{A}}_{2}+\frac{96\lambda^{3}}{175}\,\bar{\partial}^{4}\,W^{\hat{A}}_{3}+\frac{344\lambda^{2}}{35}\,\bar{\partial}^{3}\,W^{\hat{A}}_{4}+\frac{4\lambda}{15}\,\bar{\partial}^{2}\,W^{\hat{A}}_{5}+3\,\bar{\partial}\,W^{\hat{A}}_{6}\Bigg{]}(\bar{w})$
$\displaystyle+\cdots\ ,$ $\displaystyle
W^{\hat{A}}_{4}(\bar{z})\,W^{\hat{B}}_{4}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,\Bigg{[}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{2}\,\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\Bigg{[}\frac{4608\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{2}\Bigg{)}$
$\displaystyle+\frac{1536\lambda^{4}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{2}+\frac{29952\lambda^{3}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{3}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{2}\Bigg{)}$
$\displaystyle+\frac{2304\lambda^{3}}{125}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,W_{3}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{3}\Bigg{)}+\frac{8256\lambda^{2}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,W_{4}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{4}\Bigg{)}$
$\displaystyle+\frac{768\lambda^{4}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{2}+\frac{14976\lambda^{3}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{3}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{3}\,W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{2}\Bigg{)}$
$\displaystyle+\frac{1152\lambda^{3}}{125}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,W_{3}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{3}\Bigg{)}+\frac{4128\lambda^{2}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,W_{4}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{4}\Bigg{)}$
$\displaystyle+\frac{1152\lambda^{4}}{125}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{2}+\frac{22464\lambda^{3}}{125}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{3}$
$\displaystyle+\frac{96\lambda^{2}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{4}+\frac{1344\lambda}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{5}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{4}\,W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{4}\,W^{\hat{C}}_{2}\Bigg{)}$
$\displaystyle+\frac{2304\lambda^{3}}{875}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{3}\,W_{3}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{3}\Bigg{)}+\frac{8256\lambda^{2}}{175}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,W_{4}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{4}\Bigg{)}$
$\displaystyle+\frac{8\lambda}{15}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,W_{5}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{5}\Bigg{)}+6\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,W_{6}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{6}\Bigg{)}$
$\displaystyle+\frac{256\lambda^{4}}{125}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{4}\,W^{\hat{C}}_{2}+\frac{4992\lambda^{3}}{125}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{3}$
$\displaystyle+\frac{48\lambda^{2}}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{4}+\frac{672\lambda}{25}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{5}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{5}\,W_{2}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{5}\,W^{\hat{C}}_{2}\Bigg{)}$
$\displaystyle+\frac{96\lambda^{3}}{175}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{4}\,W_{3}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{4}\,W^{\hat{C}}_{3}\Bigg{)}+\frac{344\lambda^{2}}{35}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{3}\,W_{4}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{4}\Bigg{)}$
$\displaystyle+\frac{4\lambda}{15}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,W_{5}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{5}\Bigg{)}+3\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,W_{6}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{6}\Bigg{)}$
$\displaystyle+\frac{64\lambda^{4}}{175}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{5}\,W^{\hat{C}}_{2}+\frac{1248\lambda^{3}}{175}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{4}\,W^{\hat{C}}_{3}$
$\displaystyle+\frac{8\lambda^{2}}{15}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,W^{\hat{C}}_{4}+\frac{112\lambda}{15}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,W^{\hat{C}}_{5}$
$\displaystyle+\frac{3}{55}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,W^{\hat{C}}_{6}+\frac{1}{2\,\lambda}\,(-\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{7}\Bigg{]}(\bar{w})+\cdots\
,$ $\displaystyle
W_{4}(\bar{z})\,Q_{\frac{7}{2}}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,Q_{\frac{3}{2}}(\bar{w})+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\frac{4608\lambda^{4}}{5}\,\bar{\partial}\,Q_{\frac{3}{2}}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\bar{\partial}^{2}\,Q_{\frac{3}{2}}+\frac{2304\lambda^{3}}{125}\,\bar{\partial}\,Q_{\frac{5}{2}}+\frac{8256\lambda^{2}}{25}\,Q_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\bar{\partial}^{3}\,Q_{\frac{3}{2}}+\frac{1152\lambda^{3}}{125}\,\bar{\partial}^{2}\,Q_{\frac{5}{2}}+\frac{4128\lambda^{2}}{25}\,\bar{\partial}\,Q_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\bar{\partial}^{4}\,Q_{\frac{3}{2}}+\frac{2304\lambda^{3}}{875}\,\bar{\partial}^{3}\,Q_{\frac{5}{2}}+\frac{8256\lambda^{2}}{175}\,\bar{\partial}^{2}\,Q_{\frac{7}{2}}+\frac{8\lambda}{15}\,\bar{\partial}\,Q_{\frac{9}{2}}+6\,Q_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\bar{\partial}^{5}\,Q_{\frac{3}{2}}+\frac{96\lambda^{3}}{175}\,\bar{\partial}^{4}\,Q_{\frac{5}{2}}+\frac{344\lambda^{2}}{35}\,\bar{\partial}^{3}\,Q_{\frac{7}{2}}+\frac{4\lambda}{15}\,\bar{\partial}^{2}\,Q_{\frac{9}{2}}+3\,\bar{\partial}\,Q_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\cdots\ ,$ $\displaystyle
W_{4}(\bar{z})\,Q^{\hat{A}}_{\frac{7}{2}}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,Q^{\hat{A}}_{\frac{3}{2}}(\bar{w})+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\frac{4608\lambda^{4}}{5}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{3}{2}}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{2304\lambda^{3}}{125}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{8256\lambda^{2}}{25}\,Q^{\hat{A}}_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{1152\lambda^{3}}{125}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{4128\lambda^{2}}{25}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\bar{\partial}^{4}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{2304\lambda^{3}}{875}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{8256\lambda^{2}}{175}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{7}{2}}+\frac{8\lambda}{15}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{9}{2}}+6\,Q^{\hat{A}}_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\bar{\partial}^{5}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{96\lambda^{3}}{175}\,\bar{\partial}^{4}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{344\lambda^{2}}{35}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{7}{2}}+\frac{4\lambda}{15}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{9}{2}}+3\,\bar{\partial}\,Q^{\hat{A}}_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\cdots\ ,$ $\displaystyle
W^{\hat{A}}_{4}(\bar{z})\,Q_{\frac{7}{2}}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,Q^{\hat{A}}_{\frac{3}{2}}(\bar{w})+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\frac{4608\lambda^{4}}{5}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{3}{2}}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{2304\lambda^{3}}{125}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{8256\lambda^{2}}{25}\,Q^{\hat{A}}_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{1152\lambda^{3}}{125}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{4128\lambda^{2}}{25}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{7}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\bar{\partial}^{4}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{2304\lambda^{3}}{875}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{8256\lambda^{2}}{175}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{7}{2}}+\frac{8\lambda}{15}\,\bar{\partial}\,Q^{\hat{A}}_{\frac{9}{2}}+6\,Q^{\hat{A}}_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})}\,\Bigg{[}\frac{64\lambda^{4}}{25}\,\bar{\partial}^{5}\,Q^{\hat{A}}_{\frac{3}{2}}+\frac{96\lambda^{3}}{175}\,\bar{\partial}^{4}\,Q^{\hat{A}}_{\frac{5}{2}}+\frac{344\lambda^{2}}{35}\,\bar{\partial}^{3}\,Q^{\hat{A}}_{\frac{7}{2}}+\frac{4\lambda}{15}\,\bar{\partial}^{2}\,Q^{\hat{A}}_{\frac{9}{2}}+3\,\bar{\partial}\,Q^{\hat{A}}_{\frac{11}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\cdots\ ,$ $\displaystyle
W^{\hat{A}}_{4}(\bar{z})\,Q^{\hat{B}}_{\frac{7}{2}}(\bar{w})=\frac{1}{(\bar{z}-\bar{w})^{6}}\,\frac{9216\lambda^{4}}{5}\,\Bigg{[}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}Q_{\frac{3}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{\frac{3}{2}}\,\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{5}}\,\Bigg{[}\frac{4608\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,Q_{\frac{3}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{3}{2}}\Bigg{)}$
$\displaystyle-\frac{1536\lambda^{4}}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{3}{2}}-\frac{29952\lambda^{3}}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,W^{\hat{C}}_{\frac{5}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{4}}\,\Bigg{[}\frac{1536\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,Q_{\frac{3}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,Q^{\hat{C}}_{\frac{3}{2}}\Bigg{)}$
$\displaystyle+\frac{2304\lambda^{3}}{125}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,Q_{\frac{5}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{5}{2}}\Bigg{)}+\frac{8256\lambda^{2}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,Q_{\frac{7}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{\frac{7}{2}}\Bigg{)}$
$\displaystyle-\frac{768\lambda^{4}}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,Q^{\hat{C}}_{\frac{3}{2}}-\frac{14976\lambda^{3}}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{5}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{3}}\,\Bigg{[}\frac{384\lambda^{4}}{5}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{3}\,Q_{\frac{3}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,Q^{\hat{C}}_{\frac{3}{2}}\Bigg{)}$
$\displaystyle+\frac{1152\lambda^{3}}{125}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{2}\,Q_{\frac{5}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,Q^{\hat{C}}_{\frac{5}{2}}\Bigg{)}+\frac{4128\lambda^{2}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}\,Q_{\frac{7}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{7}{2}}\Bigg{)}$
$\displaystyle-\frac{1152\lambda^{4}}{125}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{3}\,Q^{\hat{C}}_{\frac{3}{2}}-\frac{22464\lambda^{3}}{125}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{2}\,Q^{\hat{C}}_{\frac{5}{2}}$
$\displaystyle-\frac{96\lambda^{2}}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}\,Q^{\hat{C}}_{\frac{7}{2}}-\frac{1344\lambda}{25}\,(\frac{i}{2})\,f^{\hat{A}\hat{B}\hat{C}}\,Q^{\hat{C}}_{\frac{9}{2}}\Bigg{]}(\bar{w})$
$\displaystyle+\frac{1}{(\bar{z}-\bar{w})^{2}}\,\Bigg{[}\frac{384\lambda^{4}}{25}\,\Bigg{(}\frac{1}{K}\,\delta^{\hat{A}\hat{B}}\,\bar{\partial}^{4}\,Q_{\frac{3}{2}}+\frac{1}{2}\,d^{\hat{A}\hat{B}\hat{C}}\,\bar{\partial}^{4}\,Q^{\hat{C}}_{\frac{3}{2}}\Bigg{)}$