Fermionic Construction in the Supersymmetric Coset Model

Changhyun Ahn†, Jaesung Hong†, and Man Hea Kim∗

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

$\ast$ Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

It is known previously that the operator product expansion (OPE) between the
first ${\cal N}=3$ multiplet and itself contains the second ${\cal N}=3$
multiplet in the supersymmetric coset model. In this paper, by using their
realizations in terms of various fermions, we compute the four kinds of OPEs
between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$
where the group of the coset contains $SU(N+M)$. By supersymmetrizing the
above OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities
between the currents, we determine the ${\cal N}=3$ supersymmetric OPE between
the first and the second ${\cal N}=3$ multiplets completely. The right hand
side of this OPE contains the various ${\cal N}=3$ multiplets: the $SO(3)$
singlet ${\cal N}=3$ multiplets of superspin-$\frac{3}{2},2,3,4$ and the
$SO(3)$ triplet ${\cal N}=3$ multiplets of
superspin-$\frac{5}{2},3,\frac{7}{2}$. The ${\cal N}=2$ superspace description
and the decoupling of the spin-$\frac{1}{2}$ current of the ${\cal N}=3$
superconformal algebra are also described.

###### Contents

  1. 1  Introduction
  2. 2  The OPEs between the first and second ${\cal N}=3$ multiplets in the component approach
    1. 2.1  The OPE between the spin-$\frac{3}{2}$ current and the spin-$2$ current
    2. 2.2  The OPE between the spin-$2$ currents and the spin-$2$ current
    3. 2.3  The OPE between the spin-$\frac{5}{2}$ currents and the spin-$2$ current
    4. 2.4  The OPE between the spin-$3$ current and the spin-$2$ current
    5. 2.5  The ${\cal N}=3$ superspace and the Jacobi identity
  3. 3  The ${\cal N}=3$ supersymmetric OPE
  4. 4  Conclusions and outlook
  5. A The ${\cal N}=2$ superspace description of ${\cal N}=3$ multiplets
  6. B The ${\cal N}=3$ multiplets after the decoupling of the spin-$\frac{1}{2}$ current of ${\cal N}=3$ superconformal algebra

##  1  Introduction

The Virasoro zeromode acting on the primary state, which corresponds to the
WZW primary field, is given by the quadratic Casimir operator of the finite
Lie algebra with some coefficient. Then the spin of the primary state is the
$\frac{1}{2}$ times the quadratic Casimir eigenvalues divided by the sum of
the level and the dual Coxeter number of the finite Lie algebra [1, 2]. The
adjoint representation at the level, which is equal to the dual Coxeter
number, has the spin-$\frac{1}{2}$. For the diagonal bosonic coset model [1],
the ${\cal N}=1$ supersymmetry generator of spin-$\frac{3}{2}$ was initiated
by [3, 4, 5] and see also the previous coset constructions in [6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16] for the various supersymmetric models 111The other way
to generalize the bosonic theory to the supersymmetric theory in two
dimensional conformal field theory is to introduce the complex fermions in the
bosonic coset model.. In the different type of ${\cal N}=2$ supersymmetric
coset model [17, 18], its ${\cal N}=3$ version in [19] has been studied.
Moreover, the lowest ${\cal N}=3$ multiplet in terms of various fermions has
been obtained in [20].

The coset model in [17, 18] is described as

$\displaystyle\frac{{SU}(N+M)_{k}\times{SO}(2NM)_{1}}{{SU}(N)_{k+M}\times{SU}(M)_{k+N}\times{U}(1)_{NM(N+M)(k+N+M)}}.$
(1.1)

Note that the level $k$ in (1.1) should be equal to the dual Coxeter number of
$SU(N+M)$ for its ${\cal N}=3$ version: $k=N+M$. The additional third
supersymmetry generator of spin-$\frac{3}{2}$ is obtained from the adjoint
fermions of $SU(N+M)$ and the fermions of $SO(2NM)_{1}$ [19].

In this paper, by using the explicit realizations for the ${\cal N}=3$
multiplets in terms of various fermions, we calculate the four kinds of OPEs
between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$
($N=3,M=2$) with the help of the Thielemans package [21] and a mathematica
[22]. By supersymmetrizing the above OPEs in ${\cal N}=3$ superspace and using
the various Jacobi identities between the currents as done in [23, 24], we
determine the ${\cal N}=3$ supersymmetric OPE between the first and the second
${\cal N}=3$ multiplets completely. The right hand side of this OPE contains
the following ${\cal N}=3$ multiplets 222 The ${\cal N}=3$ superspace
coordinates [25, 26, 27, 28, 29] are described by $(Z,\overline{Z}),$ where
$Z=(z,\theta^{i})$ and $\overline{Z}=(\bar{z},$ $\bar{\theta}^{i})$, with
$SO(3)$-vector index $i=1,2,3$. The left covariant spinor derivative is
$D^{i}=\theta^{i}\frac{\partial}{\partial
z}+\frac{\partial}{\partial{\theta^{i}}}$ satisfying the anticommutators
$\\{D^{i},D^{j}\\}=2\delta^{ij}\frac{\partial}{\partial z}$. The fermionic
coordinate difference for given index $i$ is defined as
$\theta_{12}^{i}=\theta_{1}^{i}-\theta_{2}^{i}$, and the bosonic coordinate
difference is given by $z_{12}=z_{1}-z_{2}-\theta_{1}^{i}\theta_{2}^{i}$. We
will follow the conventions used in [20].

$\displaystyle{\bar{\bf\Phi}}^{(\frac{3}{2})}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(\frac{3}{2})}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(2),i}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(\frac{5}{2}),i}(z)+\theta^{3-0}\bar{\phi}^{(3)}(z),$
$\displaystyle{\bar{\bf\Phi}}^{(2)}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(2)}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(\frac{5}{2}),i}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(3),i}(z)+\theta^{3-0}\bar{\phi}^{(\frac{7}{2})}(z),$
$\displaystyle{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(\frac{5}{2}),\alpha}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(3),i,\alpha}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(\frac{7}{2}),i,\alpha}(z)+\theta^{3-0}\bar{\phi}^{(4),\alpha}(z),$
$\displaystyle{\bar{\bf\Phi}}^{(3),\alpha}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(3),\alpha}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(\frac{7}{2}),i,\alpha}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(4),i,\alpha}(z)+\theta^{3-0}\bar{\phi}^{(\frac{9}{2}),\alpha}(z),$
$\displaystyle{\tilde{\bf\Phi}}^{(3)}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\tilde{\psi}^{(3)}(z)+\theta^{i}\frac{i}{2}\tilde{\phi}^{(\frac{7}{2}),i}(z)+\theta^{3-i}\frac{1}{2}\tilde{\psi}^{(4),i}(z)+\theta^{3-0}\tilde{\phi}^{(\frac{9}{2})}(z),$
$\displaystyle{\bar{\bf\Phi}}^{(\frac{7}{2}),\alpha}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(\frac{7}{2}),\alpha}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(4),i,\alpha}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(\frac{9}{2}),i,\alpha}(z)+\theta^{3-0}\bar{\phi}^{(5),\alpha}(z),$
$\displaystyle{\bar{\bf\Phi}}^{(4)}(Z)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\psi}^{(4)}(z)+\theta^{i}\frac{i}{2}\bar{\phi}^{(\frac{9}{2}),i}(z)+\theta^{3-i}\frac{1}{2}\bar{\psi}^{(5),i}(z)+\theta^{3-0}\bar{\phi}^{(\frac{11}{2})}(z).$
(1.2)

These ${\cal N}=3$ multiplets do not overlap with the ones in [20]. We do not
present the coset field contents for these currents because we know only for
fixed $(N,M)$ values and they are rather complicated 333The independent terms
of the spin-$4$ current $\bar{\psi}^{(4)}$ for $(N,M)=(3,2)$ are more than
$180000$..

##  2  The OPEs between the first and second ${\cal N}=3$ multiplets in the
component approach

Let us consider the four types of OPEs between the currents for fixed
$(N,M)=(3,2)$ where the central charge is $c=9$ 444The central charge is given
by $c=\frac{3}{2}NM$.. By using the ${\cal N}=3$ supersymmetry, the remaining
twelve types of OPEs can be determined. By using the Jacobi identity between
these OPEs with arbitrary coefficients, we obtain the final structure
constants in these OPEs in terms of the arbitrary central charge.

###  2.1  The OPE between the spin-$\frac{3}{2}$ current and the spin-$2$
current

In [20], the lowest components of the first and second ${\cal N}=3$ multiplet
in terms of the coset fields are known. Then we can calculate the following
OPE

$\displaystyle\psi^{(\frac{3}{2})}(z)\psi^{(2)}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{2}}\Bigg{[}-2i\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\bar{\psi}^{\left(\frac{3}{2}\right)}\Bigg{]}(w)$
(2.1) $\displaystyle+$
$\displaystyle\frac{1}{(z-w)}\Bigg{[}\frac{1}{3}\partial(\mbox{pole-2})\Bigg{]}(w)+\cdots.$

It turns out that there exists a new 555Although the field contents of this
spin-$\frac{3}{2}$ current are the same as the one spin-$\frac{3}{2}$ current
appearing in the left hand side of (2.1), each coefficient appearing in the
composite operators is different from each other. We have
$6\sqrt{\frac{3}{65}}(\frac{27}{20}\sqrt{\frac{3}{130}}(\frac{-4}{9}J_{1}^{\alpha}+J_{2}^{\alpha}+j^{\alpha})\Psi^{\alpha}(z)+\frac{13}{10\sqrt{15}}(-J_{1}^{\rho}+J_{2}^{\rho}+j^{\rho})\Psi^{\rho}(z))$
while
${\psi}^{\left(\frac{3}{2}\right)}(z)=\frac{3\sqrt{\frac{3}{13}}}{5}(-\frac{4}{9}J_{1}^{\alpha}+J_{2}^{\alpha}+j^{\alpha})\Psi^{\alpha}(z)+\frac{\sqrt{\frac{3}{2}}}{5}(-J_{1}^{\rho}+J_{2}^{\rho}+j^{\rho})\Psi^{\alpha}(z)$
from [20] where the adjoint index $\alpha$ is for $SU(N)$ and the adjoint
index $\rho$ is for $SU(M)$. The various spin-$1$ currents are defined in
Appendix $C$ of [20].  primary current $\bar{\psi}^{\left(\frac{3}{2}\right)}$
with the structure constant
$\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}$ at the
second order pole of (2.1) 666The upper index of the structure constant stands
for the spin of the ${\cal N}=3$ multiplet in (1.2). In this paper, there are
seven undetermined structure constants associated with the ones in (1.2) and
they can be fixed by calculating the OPEs between each ${\cal N}=3$ multiplet
and itself.. At the first order pole its descendant term appears. It is
straightforward to compute all the other components in the first ${\cal N}=3$
multiplet (1.2) by using Appendix $B$ (the ${\cal N}=3$ primary conditions) of
[20].

###  2.2  The OPE between the spin-$2$ currents and the spin-$2$ current

Now we can move on the next component of the first ${\cal N}=3$ multiplet and
the lowest component of the second ${\cal N}=3$ multiplet of [20] 777 We have
the ${\cal N}=3$ stress energy tensor as follows: ${\bf
J}(Z)=\frac{i}{2}\Psi(z)+\theta^{i}\frac{i}{2}J^{i}(z)+\theta^{3-i}\frac{1}{2}G^{i}(z)+\theta^{3-0}T(z)$.

$\displaystyle\phi^{(2),i}(z)\psi^{(2)}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{(z-w)^{2}}\Bigg{[}-\frac{2}{3}i\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\bar{\phi}^{(2),i}\Bigg{]}(w)+\frac{1}{(z-w)}\Bigg{[}\frac{1}{2}\partial(\mbox{pole-2})$
(2.2) $\displaystyle+$
$\displaystyle\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\left(\frac{27i(c+1)}{c(c+3)(c-3)}\Psi
J^{i}\bar{\psi}^{(\frac{3}{2})}-\frac{9i(c+1)}{(c+3)(c-3)}G^{i}\bar{\psi}^{(\frac{3}{2})}\right.$
$\displaystyle-$
$\displaystyle\left.\frac{(c-15)}{(c+3)(c-3)}\varepsilon^{ijk}J^{j}\bar{\phi}^{(2),k}+\frac{i(5c-3)}{(c+3)(c-3)}\partial\bar{\phi}^{(2),i}\right)$
$\displaystyle+$
$\displaystyle\bar{C}_{(3)(2)}^{(2)}\left(\frac{c}{10(c-3)}\bar{\psi}^{(3),i}-\frac{3}{10(c-3)}\Psi\bar{\phi}^{(\frac{5}{2}),i}-\frac{6}{5(c-3)}J^{i}\bar{\psi}^{(2)}\right)$
$\displaystyle-$
$\displaystyle\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}\left(\frac{c}{3(c-1)}\delta^{i\alpha}(T^{j})^{\alpha\beta}\bar{\phi}^{(3),j,\beta}+\frac{2}{(c-1)}\Psi\bar{\psi}^{(\frac{5}{2}),\alpha=i}\right)$
$\displaystyle-$ $\displaystyle
2i\bar{C}_{(2)(2)}^{(3)}\bar{\psi}^{(3),\alpha=i}\Bigg{]}(w)+\cdots.$

After subtracting the descendant term in the first order pole of (2.2), we are
left with three primary operators and a new primary current of spin-$3$. First
of all, it is not clear how we can consider the last two primary operators,
$\bar{C}_{(3)(2)}^{(2)}$ and
$\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}$ terms, at
the first order pole. The way we do here is to calculate the highest order
poles appearing in next two subsections, identify the two new lowest
components of the second and third ${\cal N}=3$ multiplet and obtain all the
other components in each ${\cal N}=3$ multiplet explicitly. After that, we
return to the first order pole of (2.2) by allowing us to have all the
possible composite operator terms of spin-$3$ and then it turns out that we
are left with the first order pole above.

Note that there appears the dependence on the $SO(3)$ generators
888Explicitly, we have the nonzero components of these generators as follows
[20]: $(T^{1})^{23}=-(T^{1})^{32}=-i$, $(T^{2})^{13}=-(T^{2})^{31}=i$ and
$(T^{3})^{12}=-(T^{3})^{21}=-i$ satisfying the algebra
$[T^{i},T^{j}]=i\,\epsilon^{ijk}\,T^{k}$. $T^{i}$ where $i=1,2,3$ compared
with the work of [20]. The two indices $j$ and $\beta$ coming from
$(T^{j})^{\alpha\beta}$ and $\bar{\phi}^{(3),j,\beta}$ are contracted with
each other and the index $\alpha$ becomes the $SO(3)$ vector representation
due to the free index $i$ in both sides. Although we have considered all the
possible terms by including the nonlinear terms in the ${\cal N}=3$
multiplets, the above results imply that the nonlinear terms come from the
components of ${\cal N}=3$ stress energy tensor and the components of ${\cal
N}=3$ multiplets. In other words, there are no nonlinear terms in the
components of ${\cal N}=3$ multiplets.

Therefore, we have obtained the new lowest spin-$3$ component of the fourth
${\cal N}=3$ multiplet in (1.2) and the remaining components can be determined
by using the ${\cal N}=3$ supersymmetry currents of the ${\cal N}=3$ stress
tensor as before.

###  2.3  The OPE between the spin-$\frac{5}{2}$ currents and the spin-$2$
current

From the third component of the first ${\cal N}=3$ multiplet and the lowest
component of the second ${\cal N}=3$ multiplet, we obtain the following OPE

$\displaystyle\psi^{(\frac{5}{2}),i}(z)\psi^{2}(w)=\frac{1}{(z-w)^{2}}\Bigg{[}-\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\left(\frac{i(c-15)}{3(c-3)}\bar{\psi}^{\left(\frac{5}{2}\right),i}+\frac{i(c+9)}{c(c-3)}\Psi\bar{\phi}^{(2),i}\right.$
$\displaystyle+\left.\frac{3i(5c-3)}{c(c-3)}J^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}\right)+\frac{1}{2}\bar{C}_{(3)(2)}^{(2)}\bar{\phi}^{\left(\frac{5}{2}\right),i}+2\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=i}\Bigg{]}(w)$
$\displaystyle+\frac{1}{(z-w)}\Bigg{[}\frac{3}{5}\partial(\mbox{pole-2})+\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\Bigg{(}-\frac{27i(c+1)}{c(c+3)(c-3)}\Psi
G^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}$
$\displaystyle-\frac{1}{(5c+6)(c+3)(c-3)}\varepsilon^{ijk}(\vphantom{\frac{2i\left(19c^{2}-18c-45\right)}{5}}18(c+1)(c-1)G^{j}\bar{\phi}^{(2),k}-\frac{2i\left(19c^{2}-18c-45\right)}{5}\varepsilon^{jkl}\partial\bar{\psi}^{\left(\frac{5}{2}\right),l})$
$\displaystyle-\frac{2\left(c^{2}-18c-27\right)}{(5c+6)(c+3)(c-3)}\varepsilon^{ijk}J^{j}\bar{\psi}^{\left(\frac{5}{2}\right),k}$
$\displaystyle-\frac{i}{(c+3)(c-3)(5c+6)}(\vphantom{\frac{1}{2}}c(5c-27)\partial\Psi\bar{\phi}^{(2),i}-\frac{(5c^{3}+153c^{2}+396c+216)}{5c}\partial(\Psi\bar{\phi}^{(2),i}))$
$\displaystyle-\frac{12i(5c^{3}-18c-9)}{c(c+3)(c-3)(5c+6)}(\partial
J^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}-\frac{2}{5}\partial(J^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}))+\frac{9(c+1)}{c(c+3)(c-3)}\varepsilon^{ijk}\Psi
J^{j}\bar{\phi}^{(2),k}\Bigg{)}$
$\displaystyle+\bar{C}_{(3)(2)}^{(2)}\Bigg{(}-\frac{18}{5(c-3)(2c+9)}\Psi
J^{i}\bar{\psi}^{(2)}+\frac{3}{10(c-3)}\Psi\bar{\psi}^{(3),i}$
$\displaystyle-\frac{1}{5(c-3)(2c+9)}(6(c+9)G^{i}\bar{\psi}^{(2)}-18\partial\bar{\phi}^{\left(\frac{5}{2}\right),i})-\frac{3i(2c+3)}{5(c-3)(2c+9)}\varepsilon^{ijk}J^{j}\bar{\phi}^{\left(\frac{5}{2}\right),k}\Bigg{)}$
$\displaystyle-\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}\Bigg{(}\frac{1}{(c-1)}\delta^{i\alpha}(T^{j})^{\alpha\beta}\Psi\bar{\phi}^{(3),j,\beta}+\frac{1}{7}\delta^{i\alpha}(T^{j})^{\alpha\beta}(\bar{\psi}^{\left(\frac{7}{2}\right),j,\beta}+\frac{1}{5}(T^{j})^{\beta\gamma}\partial\bar{\psi}^{\left(\frac{5}{2}\right),\gamma})\Bigg{)}$
$\displaystyle+\bar{C}_{(2)(2)}^{(3)}\Bigg{(}\frac{6i(c-1)}{(7c-6)}\delta^{i\alpha}(T^{j})^{\alpha\beta}\bar{\phi}^{\left(\frac{7}{2}\right),j,\beta}-\frac{6i}{(7c-6)}\Psi\bar{\psi}^{(3),\alpha=i}\Bigg{)}$
$\displaystyle+\frac{1}{3}\tilde{C}_{(3)(2)}^{(3)}\tilde{\phi}^{\left(\frac{7}{2}\right),i}+2\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{7}{2}\right)}\bar{\psi}^{\left(\frac{7}{2}\right),\alpha=i}\Bigg{]}(w)+\cdots.$
(2.3)

As decribed in previous section, we have found the second order pole of the
above OPE (2.3) by calculating the highest order pole of the next OPE in next
subsection and identifying the second ${\cal N}=3$ multiplet. In other words,
the second order pole consists of a primary operator, a primary
spin-$\frac{5}{2}$ current which belongs to the second ${\cal N}=3$ multiplet
and other kind of spin-$\frac{5}{2}$ currents which is the component of the
third ${\cal N}=3$ multiplet in (1.2).

At the first order pole, there are four kinds of primary operators, a primary
spin-$\frac{7}{2}$ current belonging to the fifth ${\cal N}=3$ multiplet in
(1.2) and other primary spin-$\frac{7}{2}$ currents which are the lowest
components of the sixth ${\cal N}=3$ multiplet in (1.2). Also note that in the
term of $\bar{C}_{(2)(2)}^{(3)}$ in (2.3), we observe the presence of the
fourth ${\cal N}=3$ multiplet.

###  2.4  The OPE between the spin-$3$ current and the spin-$2$ current

The final most complicated OPE can be summarized by

$\displaystyle\phi^{(3)}(z)\psi^{(2)}(w)=\frac{1}{(z-w)^{3}}\Bigg{[}-\frac{6i}{c}\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\Psi\bar{\psi}^{\left(\frac{3}{2}\right)}+\bar{C}_{(3)(2)}^{(2)}\bar{\psi}^{(2)}\Bigg{]}(w)\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle+\frac{1}{(z-w)^{2}}\Bigg{[}\frac{3}{4}\partial(\mbox{pole-3})-\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\left(\frac{2i(7c+15)}{(c-3)(5c+6)}J^{i}\bar{\phi}^{(2),i}\right.$
$\displaystyle+\left.\frac{8i(8c+3)}{(c-3)(5c+6)}(\partial\Psi\bar{\psi}^{\left(\frac{3}{2}\right)}-\frac{1}{4}\partial(\Psi\bar{\psi}^{\left(\frac{3}{2}\right)}))+\frac{i\left(c^{2}-25c-42\right)}{(c-3)(5c+6)}\bar{\phi}^{(3)}\right)$
$\displaystyle+\frac{3}{7}\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}\bar{\phi}^{(3),i,\alpha=i}+\tilde{C}_{(3)(2)}^{(3)}\tilde{\psi}^{(3)}\Bigg{]}(w)+\frac{1}{(z-w)}\Bigg{[}\frac{3}{10}\partial^{2}(\mbox{pole-3})+\frac{2}{3}\partial(\mbox{pole-2}-\frac{3}{4}\partial(\mbox{pole-3}))$
$\displaystyle+\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}\left(\frac{3i(c-12)(c+1)}{c(c-3)(c+3)^{2}}(\Psi
J^{i}\bar{\psi}^{\left(\frac{5}{2}\right),i}-\Psi
G^{i}\bar{\phi}^{\left(2\right),i})\right.$
$\displaystyle+\frac{135i(c+1)}{c(c-3)(c+3)^{2}}\Psi
J^{i}J^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}-\frac{i(c-12)(c+1)}{(c-3)(c+3)^{2}}(G^{i}\bar{\psi}^{\left(\frac{5}{2}\right),i}-2\partial\bar{\phi}^{(3)})$
$\displaystyle-\frac{1}{(c-3)(c+3)^{2}}(45i(c+1)J^{i}G^{i}\bar{\psi}^{\left(\frac{3}{2}\right)}-\frac{2ic(c+33)}{3}J^{i}\partial\bar{\phi}^{(2),i})$
$\displaystyle-\frac{i\left(4c^{2}-3c-135\right)}{3(c-3)(c+3)^{2}}\partial
J^{i}\bar{\phi}^{(2),i}-\frac{i(16c^{3}+21c^{2}+153c-108)}{2c(c+3)^{2}(c-3)}\partial^{2}\Psi\bar{\psi}^{\left(\frac{3}{2}\right)}$
$\displaystyle\left.+\frac{4i(4c^{2}+3c+63)}{3(c+3)^{2}(c-3)}\partial(\partial\Psi\bar{\psi}^{\left(\frac{3}{2}\right)})-\frac{i\left(8c^{3}-21c^{2}+423c+324\right)}{15c(c-3)(c+3)^{2}}\partial^{2}(\Psi\bar{\psi}^{\left(\frac{3}{2}\right)})\right)$
$\displaystyle+\bar{C}_{(3)(2)}^{(2)}\left(-\frac{18\left(8c^{2}+53c+57\right)}{5(c-3)(2c+9)\left(c^{2}+26c+9\right)}J^{i}J^{i}\bar{\psi}^{(2)}\right.$
$\displaystyle+\frac{3\left(2c^{3}+7c^{2}+63c+90\right)}{5(c-3)(2c+9)\left(c^{2}+26c+9\right)}J^{i}\bar{\psi}^{(3),i}+\frac{12c\left(4c^{2}+9c-99\right)}{5(c-3)(2c+9)\left(c^{2}+26c+9\right)}T\bar{\psi}^{(2)}$
$\displaystyle-\frac{9\left(6c^{3}+17c^{2}-105c-36\right)}{10(c-3)(2c+9)\left(c^{2}+26c+9\right)}\partial^{2}\bar{\psi}^{(2)}-\frac{27\left(6c^{2}+21c-1\right)}{10(c-3)(2c+9)\left(c^{2}+26c+9\right)}\Psi
J^{i}\bar{\phi}^{\left(\frac{5}{2}\right),i}$
$\displaystyle\left.+\frac{3\left(14c^{3}+49c^{2}-129c-180\right)}{10(c-3)(2c+9)\left(c^{2}+26c+9\right)}G^{i}\bar{\phi}^{\left(\frac{5}{2}\right),i}-\frac{18\left(4c^{2}+9c-99\right)}{5(c-3)(2c+9)\left(c^{2}+26c+9\right)}\partial\Psi\Psi\bar{\psi}^{(2)}\right)$
$\displaystyle+\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{5}{2}\right)}\left(-\frac{2c}{(c-1)(c+15)}\delta^{i\alpha}(T^{j})^{\alpha\beta}J^{i}\bar{\phi}^{(3),j,\beta}\right.-\frac{30}{(c-1)(c+15)}\Psi
J^{i}\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=i}$
$\displaystyle+\left.\frac{6c}{(c-1)(c+15)}(G^{i}\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=i}-\frac{1}{3}\partial\bar{\phi}^{(3),i,\alpha=i})\right)$
$\displaystyle-\bar{C}_{(2)(2)}^{(3)}\left(\frac{3ic(c-2)}{2(c+3)(7c-6)}\bar{\psi}^{(4),i,\alpha=i}+\frac{9i(c-2)}{2(c+3)(7c-6)}\Psi\bar{\phi}^{\left(\frac{7}{2}\right),i,\alpha=i}\right.$
$\displaystyle+\left.\frac{12i(4c-3)}{(c+3)(7c-6)}J^{i}\bar{\psi}^{(3),\alpha=i}\right)+\frac{4}{9}\bar{C}_{\left(\frac{5}{2}\right)(2)}^{\left(\frac{7}{2}\right)}\bar{\phi}^{(4),i,\alpha=i}+\bar{C}_{(3)(2)}^{(4)}\bar{\psi}^{(4)}\Bigg{]}(w)+\cdots.$
(2.4)

As explained before, the third order pole of (2.4) contains the lowest
component of the second ${\cal N}=3$ multiplet in (1.2). The component of
fifth ${\cal N}=3$ multiplet appears at the second order pole. We arrive at
the lowest components of the sixth and seventh ${\cal N}=3$ multiplets in the
first order pole of (2.4). The two quasiprimary operators with
$\bar{C}_{\left(\frac{3}{2}\right)(2)}^{\left(\frac{3}{2}\right)}$ and
$\bar{C}_{(3)(2)}^{(2)}$ at the first order pole appear while the next two
operators are primary.

Therefore, we have obtained the new seven ${\cal N}=3$ multiplets in (1.2) by
checking the presence of lowest components of those multiplets.

###  2.5  The ${\cal N}=3$ superspace and the Jacobi identity

Because the neccessary four kinds of OPEs in the previous subsections are
found explicitly, we can use ${\cal N}=3$ supersymmetry in the coset model we
are describing to determine the remaining OPEs between the first and the
second ${\cal N}=3$ multiplets. That is, we simply generalize the above four
kinds of OPEs to the complete OPEs in ${\cal N}=3$ superspace by the following
replacement

$\displaystyle{\psi}^{\left(\Delta\right),\alpha}$ $\displaystyle\rightarrow$
$\displaystyle-2i{\bf\Phi}^{(\Delta),\alpha},$
$\displaystyle{\phi}^{\left(\Delta+\frac{1}{2}\right),i,\alpha}$
$\displaystyle\rightarrow$ $\displaystyle-2iD^{i}{\bf\Phi}^{(\Delta),\alpha},$
$\displaystyle{\psi}^{\left(\Delta+1\right),i,\alpha}$
$\displaystyle\rightarrow$
$\displaystyle-2D^{3-i}{\bf\Phi}^{(\Delta),\alpha},$
$\displaystyle{\phi}^{\left(\Delta+\frac{3}{2}\right),\alpha}$
$\displaystyle\rightarrow$ $\displaystyle-D^{3-0}{\bf\Phi}^{(\Delta),\alpha},$
(2.5)

together with the appropriate fermionic coordinates and the distance between
the point $Z_{1}$ and the point $Z_{2}$ 999Similarly, we have, for the ${\cal
N}=3$ multiplets in (1.2),
$\bar{\psi}^{\left(\Delta\right),\alpha}\rightarrow-2i\bar{\bf\Phi}^{(\Delta),\alpha}$,
$\bar{\phi}^{\left(\Delta+\frac{1}{2}\right),i,\alpha}\rightarrow-2iD^{i}\bar{\bf\Phi}^{(\Delta),\alpha}$,
$\bar{\psi}^{\left(\Delta+1\right),i,\alpha}\rightarrow-2D^{3-i}\bar{\bf\Phi}^{(\Delta),\alpha}$
and
$\bar{\phi}^{\left(\Delta+\frac{3}{2}\right),\alpha}\rightarrow-D^{3-0}\bar{\bf\Phi}^{(\Delta),\alpha}$..

Then we can read off the remaining OPEs by taking the ${\cal N}=3$
superderivatives and putting the fermionic coordinates to zero. Because the
complete OPEs are known in terms of the ${\cal N}=3$ stress energy tensor and
the various ${\cal N}=3$ multiplets explicitly, we can put the arbitrary
coefficients in the right hand sides of the complete OPEs and the number of
unknown coefficients is given by sixty seven. By using the Jacobi identity
between the currents of the ${\cal N}=3$ stress energy tensor and the currents
of the ${\cal N}=3$ multiplets, all the structure constants are fixed in terms
of the central charge. Some of them appear in the previous OPEs in (2.1),
(2.2), (2.3) and (2.4).

Then by writing down these OPEs in ${\cal N}=3$ superspace, we have the final
form in next section. In Appenidx $A$, we present the ${\cal N}=3$ multiplets
in (1.2) by using its ${\cal N}=2$ superspace approach. In Appendix $B$, by
decoupling the spin-$\frac{1}{2}$ current of the ${\cal N}=3$ superconformal
algebra, the corresponding components of ${\cal N}=3$ multiplets are
described.

##  3  The ${\cal N}=3$ supersymmetric OPE

Therefore, by reexpressing all the component OPEs found in previous section in
${\cal N}=3$ supersymmetric way, we arrive at the final ${\cal N}=3$
supersymmetric OPE can be described as

$\displaystyle{\bf\Phi}^{\left(\frac{3}{2}\right)}(Z_{1}){\bf\Phi}^{(2)}(Z_{2})=\frac{1}{z_{12}^{2}}\Bigg{[}\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}{\bar{\bf\Phi}}^{(\frac{3}{2})}\Bigg{]}(Z_{2})+\frac{\theta_{12}^{3-0}}{z_{12}^{3}}\Bigg{[}-\frac{12}{c}\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}+\bar{C}_{(3)(2)}^{(2)}{\bar{\bf\Phi}}^{(2)}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}^{2}}\Bigg{[}\frac{1}{3}\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}\Bigg{]}(Z_{2})+\frac{\theta_{12}^{3-i}}{z_{12}^{2}}\Bigg{[}\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}\left(-\frac{(c-15)}{6(c-3)}D^{3-i}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{(c+9)}{c(c-3)}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right.$
$\displaystyle-\left.\frac{3(5c-3)}{c(c-3)}D^{i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right)+\frac{1}{4}\bar{C}_{(3)(2)}^{(2)}D^{i}{\bar{\bf\Phi}}^{(2)}+\bar{C}_{(\frac{5}{2})(2)}^{(\frac{5}{2})}\delta^{i\alpha}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{3-0}}{z_{12}^{2}}\Bigg{[}\frac{3}{4}\partial(\frac{\theta^{3-0}}{z_{12}^{3}}-\mbox{term})-\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}\left(\frac{4(7c+15)}{(c-3)(5c+6)}D^{i}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right.$
$\displaystyle\left.+\frac{16(8c+3)}{(c-3)(5c+6)}(\partial{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{1}{4}\partial({\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}))+\frac{(c^{2}-25c-42)}{2(c-3)(5c+6)}D^{3-0}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right)$
$\displaystyle+\frac{3}{7}\bar{C}_{(\frac{5}{2})(2)}^{(\frac{5}{2})}\delta^{i\alpha}D^{i}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}+\tilde{C}_{(3)(2)}^{(3)}{\tilde{\bf\Phi}}^{(3)}\Bigg{]}(Z_{2})+\frac{1}{z_{12}}\Bigg{[}\frac{1}{3}\partial(\frac{1}{z_{12}^{2}}-\mbox{term})\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{i}}{z_{12}}\Bigg{[}\frac{1}{2}\partial(\frac{\theta_{12}^{i}}{z_{12}^{2}}-\mbox{term})+\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}\left(\frac{54(c+1)}{c(c+3)(c-3)}{\bf
J}D^{i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{9(c+1)}{(c+3)(c-3)}D^{3-i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right.$
$\displaystyle-\left.\frac{(c-15)}{(c+3)(c-3)}\varepsilon^{ijk}D^{j}{\bf
J}D^{k}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{(5c-3)}{2(c+3)(c-3)}\partial{D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}}\right)$
$\displaystyle-\bar{C}_{(3)(2)}^{(2)}\left(-\frac{c}{20(c-3)}D^{3-i}{\bar{\bf\Phi}}^{(2)}+\frac{3}{10(c-3)}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(2)}+\frac{6}{5(c-3)}D^{i}{\bf
J}{\bar{\bf\Phi}}^{(2)}\right)$
$\displaystyle-\bar{C}_{(\frac{5}{2})(2)}^{(\frac{5}{2})}\left(\frac{ic}{6(c-1)}\delta^{i\alpha}(T^{j})^{\alpha\beta}D^{j}{\bar{\bf\Phi}}^{(\frac{5}{2}),\beta}+\frac{2}{(c-1)}\delta^{i\alpha}{\bf
J}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}\right)+\bar{C}_{(2)(2)}^{(3)}\delta^{i\alpha}{\bar{\bf\Phi}}^{(3),\alpha}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{3-i}}{z_{12}}\Bigg{[}\frac{3}{5}\partial(\frac{\theta^{3-i}}{z_{12}^{2}}-\mbox{term})+\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}\left(\frac{54(c+1)}{c(c+3)(c-3)}{\bf
J}D^{3-i}{\bf J}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right.$
$\displaystyle+\frac{\varepsilon^{ijk}}{(c+3)(c-3)(5c+6)}(\vphantom{\frac{1}{2}}18(c+1)(c-1)D^{3-j}{\bf
J}D^{k}{\bar{\bf\Phi}}^{(\frac{3}{2})}$
$\displaystyle+\frac{(19c^{2}-18c-45)}{5}\varepsilon^{jkl}\partial
D^{3-l}{\bar{\bf\Phi}}^{(\frac{3}{2})})+\frac{2(c^{2}-18c-27)}{(c+3)(c-3)(5c+6)}\varepsilon^{ijk}D^{j}{\bf
J}D^{3-k}{\bar{\bf\Phi}}^{(\frac{3}{2})}$
$\displaystyle-\frac{1}{(c+3)(c-3)(5c+6)}(\vphantom{\frac{1}{2}}c(5c-27)\partial{\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{(5c^{3}+153c^{2}+396c+216)}{5c}\partial({\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}))$
$\displaystyle\left.-\frac{12(5c^{3}-18c-9)}{c(c+3)(c-3)(5c+6)}(\partial
D^{i}{\bf J}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{2}{5}\partial(D^{i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}))-\frac{18(c+1)}{c(c+3)(c-3)}\varepsilon^{ijk}{\bf
J}D^{j}{\bf J}D^{k}{\bar{\bf\Phi}}^{(\frac{3}{2})}\right)$
$\displaystyle+\bar{C}_{(3)(2)}^{(2)}\left(\frac{36}{5(c-3)(2c+9)}{\bf
J}D^{i}{\bf J}{\bar{\bf\Phi}}^{(2)}-\frac{3}{10(c-3)}{\bf
J}D^{3-i}{\bar{\bf\Phi}}^{(2)}\right.$
$\displaystyle\left.-\frac{1}{5(2c+9)(c-3)}(-6(c+9)D^{3-i}{\bf
J}{\bar{\bf\Phi}}^{(2)}-9\partial
D^{i}{\bar{\bf\Phi}}^{(2)})-\frac{3(2c+3)}{5(2c+9)(c-3)}\varepsilon^{ijk}D^{j}{\bf
J}D^{k}{\bar{\bf\Phi}}^{(2)}\right)$
$\displaystyle\left.+\bar{C}_{(\frac{5}{2})(2)}^{(\frac{5}{2})}\left(\frac{i}{(c-1)}\delta^{i\alpha}(T^{j})^{\alpha\beta}{\bf
J}D^{j}{\bar{\bf\Phi}}^{(\frac{5}{2}),\beta}\right.-\frac{i}{14}\delta^{i\alpha}(T^{j})^{\alpha\beta}(-D^{3-j}{\bar{\bf\Phi}}^{(\frac{5}{2}),\beta}-\frac{i}{5}(T^{j})^{\beta\gamma}\partial{\bar{\bf\Phi}}^{(\frac{5}{2}),\gamma})\right)$
$\displaystyle+\bar{C}_{(2)(2)}^{(3)}\left(\frac{3i(c-1)}{(7c-6)}\delta^{i\alpha}(T^{j})^{\alpha\beta}D^{j}{\bar{\bf\Phi}}^{(3),\beta}-\frac{6}{(7c-6)}\delta^{i\alpha}{\bf
J}{\bar{\bf\Phi}}^{(3),\alpha}\right)$
$\displaystyle+\frac{1}{6}\tilde{C}_{(3)(2)}^{(3)}D^{i}{\tilde{\bf\Phi}}^{(3)}+\bar{C}_{(\frac{5}{2})(2)}^{(\frac{7}{2})}\delta^{i\alpha}{\bar{\bf\Phi}}^{(\frac{7}{2}),\alpha}\Bigg{]}(Z_{2})$
$\displaystyle+\frac{\theta_{12}^{3-0}}{z_{12}}\Bigg{[}\frac{3}{10}\partial^{2}(\frac{\theta^{3-0}}{z_{12}^{3}}-\mbox{term})+\frac{2}{3}\partial\left((\frac{\theta^{3-0}}{z_{12}^{2}}-\mbox{term})-\frac{3}{4}\partial(\frac{\theta^{3-0}}{z_{12}^{3}}-\mbox{term})\right)$
$\displaystyle+\bar{C}_{(\frac{3}{2})(2)}^{(\frac{3}{2})}\left(-\frac{12(c+1)(c-12)}{c(c-3)(c+3)^{2}}({\bf
J}D^{i}{\bf J}D^{3-i}{\bar{\bf\Phi}}^{(\frac{3}{2})}-{\bf J}D^{3-i}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})})\right.$
$\displaystyle-\frac{1080(c+1)}{c(c-3)(c+3)^{2}}{\bf J}D^{i}{\bf J}D^{i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}+\frac{2(c+1)(c-12)}{(c-3)(c+3)^{2}}(D^{3-i}{\bf
J}D^{3-i}{\bar{\bf\Phi}}^{(\frac{3}{2})}+\frac{1}{2}\partial
D^{3-0}{\bar{\bf\Phi}}^{(\frac{3}{2})})$
$\displaystyle-\frac{1}{(c-3)(c+3)^{2}}(-180(c+1)D^{i}{\bf J}D^{3-i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{4c(c+33)}{3}D^{i}{\bf J}\partial
D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})})$
$\displaystyle-\frac{2(4c^{2}-3c-135)}{3(c+3)^{2}(c-3)}\partial D^{i}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(\frac{3}{2})}-\frac{(16c^{3}+21c^{2}+153c-108)}{c(c+3)^{2}(c-3)}\partial^{2}{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})}$
$\displaystyle+\left.\frac{8(4c^{2}+3c+63)}{3(c+3)^{2}(c-3)}\partial(\partial{\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})})-\frac{2(8c^{3}-21c^{2}+423c+324)}{15c(c+3)^{2}(c-3)}\partial^{2}({\bf
J}{\bar{\bf\Phi}}^{(\frac{3}{2})})\right)$
$\displaystyle+\bar{C}_{(3)(2)}^{(2)}\left(\frac{72(8c^{2}+53c+57)}{5(c^{2}+26c+9)(2c+9)(c-3)}D^{i}{\bf
J}D^{i}{\bf J}{\bar{\bf\Phi}}^{(2)}\right.$
$\displaystyle-\frac{6(2c^{3}+7c^{2}+63c+90)}{5(c^{2}+26c+9)(2c+9)(c-3)}D^{i}{\bf
J}D^{3-i}{\bar{\bf\Phi}}^{(2)}$
$\displaystyle-\frac{12c(4c^{2}+9c-99)}{5(c^{2}+26c+9)(2c+9)(c-3)}D^{3-0}{\bf
J}{\bar{\bf\Phi}}^{(2)}-\frac{9(6c^{3}+17c^{2}-105c-36)}{10(c^{2}+26c+9)(2c+9)(c-3)}\partial^{2}{\bar{\bf\Phi}}^{(2)}$
$\displaystyle+\frac{54(6c^{2}+21c-1)}{5(c^{2}+26c+9)(2c+9)(c-3)}{\bf
J}D^{i}{\bf J}D^{i}{\bar{\bf\Phi}}^{(2)}$
$\displaystyle\left.-\frac{3(14c^{3}+49c^{2}-129c-180)}{5(c^{2}+26c+9)(2c+9)(c-3)}D^{3-i}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(2)}+\frac{72(4c^{2}+9c-99)}{5(c^{2}+26c+9)(2c+9)(c-3)}\partial{\bf
J}{\bf J}{\bar{\bf\Phi}}^{(2)}\right)$
$\displaystyle+\bar{C}_{(\frac{5}{2})(2)}^{(\frac{5}{2})}\left(\frac{4ic}{(c+15)(c-1)}\delta^{i\alpha}(T^{j})^{\alpha\beta}D^{i}{\bf
J}D^{j}{\bar{\bf\Phi}}^{(\frac{5}{2}),\beta}\right.+\frac{120}{(c+15)(c-1)}\delta^{i\alpha}{\bf
J}D^{i}{\bf J}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}$
$\displaystyle+\left.\delta^{i\alpha}\frac{12c}{(c+15)(c-1)}(-D^{3-i}{\bf
J}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha}-\frac{1}{6}\partial
D^{i}{\bar{\bf\Phi}}^{(\frac{5}{2}),\alpha})\right)$
$\displaystyle+\bar{C}_{(2)(2)}^{(3)}\left(-\frac{3c(c-2)}{2(c+3)(7c-6)}\delta^{i\alpha}D^{3-i}{\bar{\bf\Phi}}^{(3),\alpha}\right.$
$\displaystyle-\left.\frac{9(c-2)}{(c+3)(7c-6)}\delta^{i\alpha}{\bf
J}D^{i}{\bar{\bf\Phi}}^{(3),\alpha}-\frac{24(4c-3)}{(c+3)(7c-6)}\delta^{i\alpha}D^{i}{\bf
J}{\bar{\bf\Phi}}^{(3),\alpha}\right)$
$\displaystyle+\frac{4}{9}\bar{C}_{(\frac{5}{2})(2)}^{(\frac{7}{2})}\delta^{i\alpha}D^{i}{\bar{\bf\Phi}}^{(\frac{7}{2}),\alpha}+\bar{C}_{(3)(2)}^{(4)}{\bar{\bf\Phi}}^{(4)}\Bigg{]}(Z_{2})+\cdots.$
(3.1)

It is straightforward to obtain all the component OPEs from (3.1) by applying
the above superderivatives to both sides and putting the fermionic coordinates
to zero (2.5). The previous nine pole terms in (2.1), (2.2), (2.3) and (2.4)
are distributed in (3.1). Compared with the result in [20], the presence of
the $SO(3)$ generators arise nontrivially. Under the large $c$ limit, most of
the terms in (3.1) survive and this classical algebra should provide the
asymptotic symmetry algebra of the matrix extension of $AdS_{3}$ bulk theory
[19, 30, 31].

##  4  Conclusions and outlook

We have obtained the nontrivial ${\cal N}=3$ supersymmetric OPE described in
(3.1) and the right hand side of this OPE contain the various ${\cal N}=3$
multiplets summarized by (1.2) where we have the explicit forms for the coset
fields with fixed $(N,M)$.

Although the vacuum character in the ${\cal N}=3$ Kazama-Suzuki model will be
found explicitly for generic $N$ and $M$, it will not be easy to observe the
extra higher spin currents completely as we add them in the right hand side of
(3.1). In order to determine the structure constants appearing in these extra
higher spin currents, we should use the other Jacobi identities between them.
If the extra higher spin currents are present in the right hand side of (3.1)
in the large $(N,M)$ values, we expect that they should appear linearly. The
structure constants in them should contain the factors $(c-6)$ corresponding
to $(N,M)=(2,2)$ case, $(c-9)$ corresponding to $(N,M)=(3,2)$ case and so on.
We may try to calculate the OPEs manually explicitly with arbitrary $(N,M)$
dependence but this is beyond the scope of this paper.

The immediate question is that in order to see the full structure of this
algebra we should compute the OPEs between the two lowest ${\cal N}=3$
multiplets, the one in [20] and the one in this paper. In other words, we
should calculate the OPEs
${\bf\Phi}^{\left(\frac{3}{2}\right)}(Z_{1})\bar{\bf\Phi}^{(\frac{3}{2})}(Z_{2})$
and
$\bar{\bf\Phi}^{\left(\frac{3}{2}\right)}(Z_{1})\bar{\bf\Phi}^{(\frac{3}{2})}(Z_{2})$
further. We expect that the ${\cal N}=3$ multiplets found in [20] will appear
in the right hand sides of above OPEs.

We can apply the fermionic construction of this paper to other coset model. If
we do not have the $SO(2MN)_{1}$ and $SU(M)_{k+N}$ in the (1.1) with some
change in the levels of the remaining factors, then we have the bosonic coset
model. Recently, some of the nontrivial OPE in this model has been found in
[32]. It would be interesting to study whether we can construct the ${\cal
N}=1$ supersymmetric coset model at the level $k=N+M$.

According to the classification of [17, 18], there is an orthogonal type of
coset model. It is an open problem to describe whether we can observe its
supersymmetric enhancement by taking the critical level as we do in this
paper.

Acknowledgments

This work of CA and JH was supported by the National Research Foundation of
Korea(NRF) grant funded by the Korea government(MSIT)(No. 2020R1F1A1066893).
MHK was supported by an appointment to the YST Program at the APCTP through
the Science and Technology Promotion Fund and Lottery Fund of the Korean
Government. MHK was also supported by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang City.

##  Appendix A The ${\cal N}=2$ superspace description of ${\cal N}=3$
multiplets

We present each ${\cal N}=3$ multiplet (1.2) in terms of two ${\cal N}=2$
multiplets as follows 101010We have $\theta=\theta^{1}-i\theta^{2}$ and
$\bar{\theta}=-\theta^{1}-i\theta^{2}$. By combining the two ${\cal N}=2$
multiplet and the third fermionic coordinate $\theta^{3}$, we can reexpress
them in terms of the ${\cal N}=3$ multiplet as in [20].:

$\displaystyle{\bar{\bf W}}^{\left(\frac{3}{2}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\psi}^{\left(\frac{3}{2}\right)}(z)+\theta(\bar{\phi}^{(2),1}+i\bar{\phi}^{(2),2})(z)+\bar{\theta}(-\bar{\phi}^{(2),1}+i\bar{\phi}^{(2),2})(z)+\theta\bar{\theta}\bar{\psi}^{\left(\frac{5}{2}\right),3}(z),$
$\displaystyle{\bar{\bf W}}^{\left(2\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\phi}^{\left(2\right),3}(z)+\theta(-\bar{\psi}^{\left(\frac{5}{2}\right),1}-i\bar{\psi}^{\left(\frac{5}{2}\right),2})(z)+\bar{\theta}(-\bar{\psi}^{\left(\frac{5}{2}\right),1}+i\bar{\psi}^{\left(\frac{5}{2}\right),2})(z)+\theta\bar{\theta}2\bar{\phi}^{\left(3\right)}(z),$
$\displaystyle{\bar{\bf W}}^{\left(2^{\prime}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\psi}^{\left(2\right)}(z)+\theta(\bar{\phi}^{\left(\frac{5}{2}\right),1}+i\bar{\phi}^{\left(\frac{5}{2}\right),2})(z)+\bar{\theta}(-\bar{\phi}^{\left(\frac{5}{2}\right),1}+i\bar{\phi}^{\left(\frac{5}{2}\right),2})(z)+\theta\bar{\theta}\bar{\psi}^{\left(3\right),3}(z),$
$\displaystyle{\bar{\bf W}}^{\left(\frac{5}{2}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\phi}^{\left(\frac{5}{2}\right),3}(z)+\theta(-\bar{\psi}^{\left(3\right),1}-i\bar{\psi}^{\left(3\right),2})(z)+\bar{\theta}(-\bar{\psi}^{\left(3\right),1}+i\bar{\psi}^{\left(3\right),2})(z)+\theta\bar{\theta}2\bar{\phi}^{\left(\frac{7}{2}\right)}(z),$
$\displaystyle{\bar{\bf W}}^{\left(\frac{5}{2}^{\prime}\right),\alpha}(Z)$
$\displaystyle=$ $\displaystyle
2\bar{\psi}^{\left(\frac{5}{2}\right),\alpha}(z)+\theta(\bar{\phi}^{(3),1,\alpha}+i\bar{\phi}^{(3),2,\alpha})(z)+\bar{\theta}(-\bar{\phi}^{(3),1,\alpha}+i\bar{\phi}^{(3),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}\bar{\psi}^{\left(\frac{7}{2}\right),3,\alpha}(z),$
$\displaystyle{\bar{\bf W}}^{\left(3\right),\alpha}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\phi}^{\left(3\right),3,\alpha}(z)+\theta(-\bar{\psi}^{\left(\frac{7}{2}\right),1,\alpha}-i\bar{\psi}^{\left(\frac{7}{2}\right),2,\alpha})(z)+\bar{\theta}(-\bar{\psi}^{\left(\frac{7}{2}\right),1,\alpha}+i\bar{\psi}^{\left(\frac{7}{2}\right),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}2\bar{\phi}^{\left(4\right),\alpha}(z),$
$\displaystyle{\bar{\bf W}}^{\left(3^{\prime}\right),\alpha}(Z)$
$\displaystyle=$ $\displaystyle
2\bar{\psi}^{\left(3\right),\alpha}(z)+\theta(\bar{\phi}^{\left(\frac{7}{2}\right),1,\alpha}+i\bar{\phi}^{\left(\frac{7}{2}\right),2,\alpha})(z)+\bar{\theta}(-\bar{\phi}^{\left(\frac{7}{2}\right),1,\alpha}+i\bar{\phi}^{\left(\frac{7}{2}\right),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}\bar{\psi}^{\left(4\right),3,\alpha}(z),$
$\displaystyle{\bar{\bf W}}^{\left(\frac{7}{2}\right),\alpha}(Z)$
$\displaystyle=$ $\displaystyle
2\bar{\phi}^{\left(\frac{7}{2}\right),3,\alpha}(z)+\theta.(-\bar{\psi}^{\left(4\right),1,\alpha}-i\bar{\psi}^{\left(4\right),2,\alpha})(z)+\bar{\theta}(-\bar{\psi}^{\left(4\right),1,\alpha}+i\bar{\psi}^{\left(4\right),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}2\bar{\phi}^{\left(\frac{9}{2}\right),\alpha}(z),$
$\displaystyle{\tilde{\bf W}}^{\left(3^{\prime}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\tilde{\psi}^{\left(3\right)}(z)+\theta(\tilde{\phi}^{\left(\frac{7}{2}\right),1}+i\tilde{\phi}^{\left(\frac{7}{2}\right),2})(z)+\tilde{\theta}(-\bar{\phi}^{\left(\frac{7}{2}\right),1}+i\tilde{\phi}^{\left(\frac{7}{2}\right),2})(z)+\theta\bar{\theta}\tilde{\psi}^{\left(4\right),3}(z),$
$\displaystyle{\tilde{\bf W}}^{\left(\frac{7}{2}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\tilde{\phi}^{\left(\frac{7}{2}\right),3}(z)+\theta(-\tilde{\psi}^{\left(4\right),1}-i\tilde{\psi}^{\left(4\right),2})(z)+\tilde{\theta}(-\bar{\psi}^{\left(4\right),1}+i\tilde{\psi}^{\left(4\right),2})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}2\tilde{\phi}^{\left(\frac{9}{2}\right)}(z),$
$\displaystyle{\bar{\bf W}}^{\left(\frac{7}{2}^{\prime}\right),\alpha}(Z)$
$\displaystyle=$ $\displaystyle
2\bar{\psi}^{\left(\frac{7}{2}\right),\alpha}(z)+\theta(\bar{\phi}^{(4),1,\alpha}+i\bar{\phi}^{(4),2,\alpha})(z)+\bar{\theta}(-\bar{\phi}^{(4),1,\alpha}+i\bar{\phi}^{(4),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}\bar{\psi}^{\left(\frac{9}{2}\right),3,\alpha}(z),$
$\displaystyle{\bar{\bf W}}^{\left(4\right),\alpha}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\phi}^{\left(4\right),3,\alpha}(z)+\theta(-\bar{\psi}^{\left(\frac{9}{2}\right),1,\alpha}-i\bar{\psi}^{\left(\frac{9}{2}\right),2,\alpha})(z)+\bar{\theta}(-\bar{\psi}^{\left(\frac{9}{2}\right),1,\alpha}+i\bar{\psi}^{\left(\frac{9}{2}\right),2,\alpha})(z)$
$\displaystyle+$
$\displaystyle\theta\bar{\theta}2\bar{\phi}^{\left(5\right),\alpha}(z),$
$\displaystyle{\bar{\bf W}}^{\left(4^{\prime}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\psi}^{\left(4\right)}(z)+\theta(\bar{\phi}^{\left(\frac{9}{2}\right),1}+i\bar{\phi}^{\left(\frac{9}{2}\right),2})(z)+\bar{\theta}(-\bar{\phi}^{\left(\frac{9}{2}\right),1}+i\bar{\phi}^{\left(\frac{9}{2}\right),2})(z)+\theta\bar{\theta}\bar{\psi}^{\left(5\right),3}(z),$
$\displaystyle{\bar{\bf W}}^{\left(\frac{9}{2}\right)}(Z)$ $\displaystyle=$
$\displaystyle
2\bar{\phi}^{\left(\frac{9}{2}\right),3}(z)+\theta(-\bar{\psi}^{\left(5\right),1}-i\bar{\psi}^{\left(5\right),2})(z)+\bar{\theta}(-\bar{\psi}^{\left(5\right),1}+i\bar{\psi}^{\left(5\right),2})(z)+\theta\bar{\theta}2\bar{\phi}^{\left(\frac{11}{2}\right)}(z).$

The various OPEs in ${\cal N}=2$ superspace can be obtained by using the
package of [33] from the component results. Or we can express (3.1) in terms
of four OPEs in ${\cal N}=2$ superspace.

##  Appendix B The ${\cal N}=3$ multiplets after the decoupling of the
spin-$\frac{1}{2}$ current of ${\cal N}=3$ superconformal algebra

We can decouple the spin-$\frac{1}{2}$ current $\Psi$ of ${\cal N}=3$
superconformal algebra and by using the following transformation of the
currents we obtain the corresponding algebra explicitly

$\displaystyle{\bar{\psi}}^{\left(\frac{3}{2}\right)}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{3}{2}\right)},\qquad{\bar{\phi}}^{\left(2\right),i}\rightarrow\bar{\phi}^{\left(2\right),i},$
$\displaystyle{\bar{\psi}}^{\left(\frac{5}{2}\right),i}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{5}{2}\right),i}-\frac{3}{c}\Psi\bar{\phi}^{\left(2\right),i},\qquad{\bar{\phi}}^{\left(3\right)}\rightarrow\bar{\phi}^{\left(3\right)}+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(\frac{3}{2}\right)}-\frac{9}{2c}\partial\Psi\bar{\psi}^{\left(\frac{3}{2}\right)},$
$\displaystyle{\bar{\psi}}^{\left(2\right)}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(2\right)},\qquad{\bar{\phi}}^{\left(\frac{5}{2}\right),i}\rightarrow\bar{\phi}^{\left(\frac{5}{2}\right),i},$
$\displaystyle{\bar{\psi}}^{\left(3\right),i}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(3\right),i}-\frac{3}{c}\Psi\bar{\phi}^{\left(\frac{5}{2}\right),i},\qquad{\bar{\phi}}^{\left(\frac{7}{2}\right)}\rightarrow\bar{\phi}^{\left(\frac{7}{2}\right)}+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(2\right)}-\frac{6}{c}\partial\Psi\bar{\psi}^{\left(2\right)},$
$\displaystyle{\bar{\psi}}^{\left(\frac{5}{2}\right),\alpha}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{5}{2}\right),\alpha},\qquad{\bar{\phi}}^{\left(3\right),i,\alpha=j}\rightarrow\bar{\phi}^{\left(3\right),i,\alpha=j}-\frac{3}{c}i\varepsilon^{ijk}\Phi\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=k},$
$\displaystyle{\bar{\psi}}^{\left(\frac{7}{2}\right),i,\alpha=j}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{7}{2}\right),i,\alpha=j}-\frac{3}{c}\Psi\left(\bar{\phi}^{\left(3\right),i,\alpha=j}+\bar{\phi}^{\left(3\right),j,\alpha=i}-\delta^{ij}\bar{\phi}^{\left(3\right),k,\alpha=k}\right),$
$\displaystyle{\bar{\phi}}^{\left(4\right),\alpha=i}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\phi}^{\left(4\right),\alpha=i}+\frac{3}{2c}i\varepsilon^{ijk}\Psi\bar{\psi}^{\left(\frac{7}{2}\right),j,\alpha=k}(z)+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=i}-\frac{15}{2c}\partial\Psi\bar{\psi}^{\left(\frac{5}{2}\right),\alpha=i},$
$\displaystyle{\bar{\psi}}^{\left(3\right),\alpha}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(3\right),\alpha},\qquad{\bar{\phi}}^{\left(\frac{7}{2}\right),i,\alpha=j}\rightarrow\bar{\phi}^{\left(\frac{7}{2}\right),i,\alpha=j}-\frac{3}{c}i\varepsilon^{ijk}\Phi\bar{\psi}^{\left(3\right),\alpha=k},$
$\displaystyle{\bar{\psi}}^{\left(4\right),i,\alpha=j}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(4\right),i,\alpha=j}-\frac{3}{c}\Psi\left(\bar{\phi}^{\left(\frac{7}{2}\right),i,\alpha=j}+\bar{\phi}^{\left(\frac{7}{2}\right),j,\alpha=i}-\delta^{ij}\bar{\phi}^{\left(\frac{7}{2}\right),k,\alpha=k}\right),$
$\displaystyle{\bar{\phi}}^{\left(\frac{9}{2}\right),\alpha=i}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\phi}^{\left(\frac{9}{2}\right),\alpha=i}+\frac{3}{2c}i\varepsilon^{ijk}\Psi\bar{\psi}^{\left(4\right),j,\alpha=k}+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(3\right),\alpha=i}-\frac{9}{c}\partial\Psi\bar{\psi}^{\left(3\right),\alpha=i},$
$\displaystyle{\tilde{\psi}}^{\left(3\right)}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\psi}^{\left(3\right)},\qquad{\tilde{\phi}}^{\left(\frac{7}{2}\right),i}\rightarrow\tilde{\phi}^{\left(\frac{7}{2}\right),i},$
$\displaystyle{\tilde{\psi}}^{\left(4\right),i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\psi}^{\left(4\right),i}-\frac{3}{c}\Psi\tilde{\phi}^{\left(\frac{7}{2}\right),i},\qquad{\tilde{\phi}}^{\left(\frac{9}{2}\right)}\rightarrow\tilde{\phi}^{\left(\frac{9}{2}\right)}+\frac{3}{2c}\Psi\partial\tilde{\psi}^{\left(3\right)}-\frac{9}{c}\partial\Psi\tilde{\psi}^{\left(3\right)},$
$\displaystyle{\bar{\psi}}^{\left(\frac{7}{2}\right),\alpha}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{7}{2}\right),\alpha},\qquad{\bar{\phi}}^{\left(4\right),i,\alpha=j}\rightarrow\bar{\phi}^{\left(4\right),i,\alpha=j}-\frac{3}{c}i\varepsilon^{ijk}\Phi\bar{\psi}^{\left(\frac{7}{2}\right),\alpha=k},$
$\displaystyle{\bar{\psi}}^{\left(\frac{9}{2}\right),i,\alpha=j}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(\frac{9}{2}\right),i,\alpha=j}-\frac{3}{c}\Psi\left(\bar{\phi}^{\left(4\right),i,\alpha=j}+\bar{\phi}^{\left(4\right),j,\alpha=i}-\delta^{ij}\bar{\phi}^{\left(4\right),k,\alpha=k}\right),$
$\displaystyle{\bar{\phi}}^{\left(5\right),\alpha=i}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\phi}^{\left(5\right),\alpha=i}+\frac{3}{2c}i\varepsilon^{ijk}\Psi\bar{\psi}^{\left(\frac{9}{2}\right),j,\alpha=k}+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(\frac{7}{2}\right),\alpha=i}-\frac{21}{2c}\partial\Psi\bar{\psi}^{\left(\frac{7}{2}\right),\alpha=i},$
$\displaystyle{\bar{\psi}}^{\left(4\right)}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(4\right)},\qquad{\bar{\phi}}^{\left(\frac{9}{2}\right),i}\rightarrow\bar{\phi}^{\left(\frac{9}{2}\right),i},$
$\displaystyle{\bar{\psi}}^{\left(5\right),i}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\psi}^{\left(5\right),i}-\frac{3}{c}\Psi\bar{\phi}^{\left(\frac{9}{2}\right),i},\qquad{\bar{\phi}}^{\left(\frac{9}{2}\right)}\rightarrow\bar{\phi}^{\left(\frac{9}{2}\right)}+\frac{3}{2c}\Psi\partial\bar{\psi}^{\left(4\right)}(z)-\frac{12}{c}\partial\Psi\bar{\psi}^{\left(4\right)}.$

We expect that the result of this paper with [20] will produce the extension
of the previous work in [34, 35].

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