A Charged and Neutral Spin-$4$ Currents in the Grassmannian-like Coset Model

By calculating the second order pole in the operator product expansion (OPE) between the charged spin-$3$ current and the neutral spin-$3$ current, we determine the primary charged spin-$4$ current. Similarly by computing the second order pole in the OPE between the neutral spin-$3$ current and itself, we obtain the primary neutral spin-$4$ current. We determine the OPE between the charged spin-$2$ current and the charged spin-$3$ current for generic parameters and the infinity limit of the level $k$ (one of the parameters) is also given for this OPE. In particular, the above primary charged spin-$4$ current occurs at the first order pole of this OPE for generic parameters. We also check that the above primary charged and neutral spin-$4$ currents occur at the second order pole of the OPE between the charged spin-$3$ current and itself for fixed parameters.