Higher spin currents with arbitrary N in the $\mathcal{N}$ = 1 stringy coset minimal model

Abstract

In the $\mathcal{N}$ = 1 supersymmetric coset model based on $\left( {A_{N-1}^{(1)}\oplus A_{N-1}^{(1) },A_{N-1}^{(1) }} \right)$ at level (k, N), the lowest $\mathcal{N}$ = 1 higher spin supercurrent with $\mathrm{spins}-\left( {\frac{5}{2},3} \right)$, in terms of two independent numerator WZW currents, is reviewed. By calculating the operator product expansions (OPE) between this $\mathcal{N}$ = 1 higher spin supercurrent and itself, the next two $\mathcal{N}$ =1 higher spin supercurrents can be generated with $\mathrm{spins}-\left( {\frac{7}{2},4} \right)$ and $\left( {4,\frac{9}{2}} \right)$. These four currents are polynomials of degree 3, 4, 4, 4 in the first numerator WZW currents with level k. The complete nonlinear OPE of the lowest $\mathcal{N}$ = 1 higher spin supercurrent for general N is obtained. The three-point functions involving two scalar primaries with one spin-2 current or spin-3 current are calculated in the large N limit for all values of the ’t Hooft coupling. In particular, the light states that appeared in the case when the second level was fixed by 1 are no longer light ones because the eigenvalues are finite in the large N limit.