Higher spin currents in the enhanced N = 3 $$ mathcal{N}=3 $$ Kazama-Suzuki model

The $$ mathcal{n}=3 $$ kazama-suzuki model at the ‘critical’ level has been found by creutzig, hikida and ronne. we construct the lowest higher spin currents of spins $$ left(frac{3}{2},2,2,2,frac{5}{2},frac{5}{2},frac{5}{2},3right) $$ in terms of various fermions. in order to obtain the operator product expansions (opes) between these higher spin currents, we describe three $$ mathcal{n}=2 $$ opes between the two $$ mathcal{n}=2 $$ higher spin currents denoted by $$ left(frac{3}{2},2,2,frac{5}{2}right) $$ and $$ left(2,frac{5}{2},frac{5}{2},3right) $$ (corresponding 36 opes in the component approach). using the various jacobi identities, the coefficient functions appearing on the right hand side of these $$ mathcal{n}=2 $$ opes are deter-mined in terms of central charge completely. then we describe them as one single $$ mathcal{n}=3 $$ ope in the $$ mathcal{n}=3 $$ superspace. the right hand side of this $$ mathcal{n}=3 $$ ope contains the so(3)-singlet $$ mathcal{n}=3 $$ higher spin multiplet of spins $$ left(2,frac{5}{2},frac{5}{2},frac{5}{2},3,3,3,frac{7}{2}right) $$ , the so(3)-singlet $$ mathcal{n}=3 $$ higher spin multiplet of spins $$ left(frac{5}{2},3,3,3,frac{7}{2},frac{7}{2},frac{7}{2},4right) $$ , and the so(3)-triplet $$ mathcal{n}=3 $$ higher spin multiplets where each multiplet has the spins $$ left(3,frac{7}{2},frac{7}{2},frac{7}{2},4,4,4,frac{9}{2}right) $$ , in addition to $$ mathcal{n}=3 $$ superconformal family of the identity operator. finally, by factoring out the $$ mathrm{spin}-frac{1}{2} $$ current of $$ mathcal{n}=3 $$ linear superconformal algebra generated by eight currents of spins $$ left(frac{1}{2},1,1,1,frac{3}{2},frac{3}{2},frac{3}{2},2right) $$ , we obtain the extension of so-called so (3) nonlinear knizhnik bershadsky algebra.