Abstract
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 \( \left(\frac{3}{2},2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3\right) \) 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 \( \left(2,\frac{5}{2},\frac{5}{2},3\right) \) (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 \( \left(\frac{5}{2},3,3,3,\frac{7}{2},\frac{7}{2},\frac{7}{2},4\right) \), 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 \( \left(\frac{1}{2},1,1,1,\frac{3}{2},\frac{3}{2},\frac{3}{2},2\right) \), we obtain the extension of so-called SO (3) nonlinear Knizhnik Bershadsky algebra.
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Ahn, C., Kim, H. Higher spin currents in the enhanced \( \mathcal{N}=3 \) Kazama-Suzuki model. J. High Energ. Phys. 2016, 1 (2016). https://doi.org/10.1007/JHEP12(2016)001
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DOI: https://doi.org/10.1007/JHEP12(2016)001