The Grassmannian-like coset model and the higher spin currents

Abstract

Abstract In the Grassmannian-like coset model, $$ \frac{\mathrm{SU}{\left(N+M\right)}_k}{\mathrm{SU}{(N)}_k\times \mathrm{U}{(1)}_{kNM\left(N+M\right)}} $$ SU N + M k SU N k × U 1 kNM N + M , Creutzig and Hikida have found the charged spin-2, 3 currents and the neutral spin-2, 3 currents previously. In this paper, as an extension of Gaberdiel-Gopakumar conjecture found ten years ago, we calculate the operator product expansion (OPE) between the charged spin-2 current and itself, the OPE between the charged spin-2 current and the charged spin-3 current and the OPE between the neutral spin-3 current and itself for generic N, M and k. From the second OPE, we obtain the new charged quasi primary spin-4 current while from the last one, the new neutral primary spin-4 current is found implicitly. The infinity limit of k in the structure constants of the OPEs is described in the context of asymptotic symmetry of MM matrix generalization of AdS3 higher spin theory. Moreover, the OPE between the charged spin-3 current and itself is determined for fixed (N, M) = (5, 4) with arbitrary k up to the third order pole. We also obtain the OPEs between charged spin-1, 2, 3 currents and neutral spin-3 current. From the last OPE, we realize that there exists the presence of the above charged quasi primary spin-4 current in the second order pole for fixed (N, M) = (5, 4). We comment on the complex free fermion realization.