In the coset model $(D_N^{ (1)} \oplus D_N^{ (1)},D_N^{ (1)})$ at levels $(k_1,k_2)$, the higher spin $4$ current that contains the quartic WZW currents contracted with completely symmetric $SO(2N)$ invariant $d$ tensor of rank $4$ is obtained. The three-point functions with two scalars are obtained for any finite $N$ and $k_2$ with $k_1=1$. They are determined also in the large $N$ 't Hooft limit. When one of the levels is the dual Coxeter number of $SO(2N)$, $k_1=2N-2$, the higher spin $\frac{ 7}{ 2}$ current, which contains the septic adjoint fermions contracted with the above $d$ tensor and the triple product of structure constants, is obtained from the operator product expansion (OPE) between the spin $\frac{ 3}{ 2}$ current living in the ${ \cal N}=1$ superconformal algebra and the above higher spin $4$ current. The OPEs between the higher spin $\frac{ 7}{ 2}, 4$ currents are described. For $k_1=k_2=2N-2$ where both levels are equal to the dual Coxeter number of $SO(2N)$, the higher spin $3$ current of $U(1)$ charge $\frac{ 4}{ 3}$, which contains the six product of spin $\frac{ 1}{ 2}$ (two) adjoint fermions contracted with the product of $d$ tensor and two structure constants, is obtained. The corresponding ${ \cal N}=2$ higher spin multiplet is determined by calculating the remaining higher spin $\frac{ 7}{ 2}, \frac{ 7}{ 2}, 4$ currents with the help of two spin $\frac{ 3}{ 2}$ currents in the ${ \cal N}=2$ superconformal algebra. The other ${ \cal N}=2$ higher spin multiplet, whose $U(1)$ charge is opposite to the one of above ${ \cal N}=2$ higher spin multiplet, is obtained. The OPE between these two ${ \cal N}=2$ higher spin mutiplets is also discussed.
Categories: hep-th